Coupled Electromagnetic Field/Circuit Simulation: Modeling ... · g angige Methode in der Praxis...

Coupled Electromagnetic Field/Circuit Simulation:Modeling and Numerical Analysis

Inaugural-Dissertation

zur

Erlangung des Doktorgrades

der Mathematisch-Naturwissenschaftlichen Fakultat

der Universitat zu Koln

vorgelegt von

Sascha Baumanns

aus Kamp-Lintfort

Berichterstatter/in: Prof. Dr. Caren TischendorfProf. Dr. Ulrich Trottenberg

Tag der mundlichen Prufung: 19.06.2012

Kurzzusammenfassung

Heutige Schaltungsmodelle verlieren in der Schaltungssimulation aufgrund der rasantentechnologischen Entwicklung, Miniaturisierung und hoherer Komplexitat von integri-erten Schaltungen zunehmend ihre Gultigkeit. Dies motiviert die direkte Kombinationvon Schaltungssimulation mit Bauelementesimulation fur kritische Schaltungsteile.In dieser Arbeit betrachten wir ein Modell von partiellen Differentialgleichungen furelektromagnetische Bauelemente - modelliert durch die Maxwell-Gleichungen - gekop-pelt mit differential-algebraischen Gleichungen, welche die einfachen Schaltungselementeeinschließlich Memristoren und die Topologie der Schaltung beschreiben.Wir untersuchen das gekoppelte System nach Diskretisierung der Maxwell-Gleichungenin einer Potentialformulierung im Ort durch die Finite Integration Technik, die einegangige Methode in der Praxis ist. Das ortsdiskretisierte gekoppelte System ist alsdifferential-algebraische Gleichung mit einem proper formulierten Hauptterm modelliert.Es werden topologische Bedingungen sowie Modellierungsbedingungen, die sicherstellen,dass der Index der differential-algebraischen Gleichung nicht großer als zwei ist, prasen-tiert. Es zeigt sich, dass der Index abhangig von der gewahlten Eichbedingung fur dieMaxwell-Gleichungen ist.Fur die erfolgreiche numerische Integration von differential-algebraischen Gleichungenspielt die Index-Charakterisierung eine entscheidende Rolle. Der Index kann als Maß furdie Empfindlichkeit der Gleichung gegenuber Storungen der Eingangsfunktionen undnumerischer Schwierigkeiten, wie der Berechnung von konsistenten Anfangswerten furZeitintegration, gesehen werden.Wir verallgemeinern Indexreduktionstechniken fur den Traktabilitatsindex fur eine all-gemeine Klasse von differential-algebraischen Gleichungen. Mit Hilfe der Indexreduktionerhalten wir lokale Losbarkeits- und Storungsaussagen fur differential-algebraische Gle-ichungen mit einem proper formulierten Hauptterm vom Index-2, und wir geben einenAlgorithmus an, um konsistente Initialisierungen fur das ortsdiskretisierte gekoppelteSystem zu bestimmen.Schließlich werden die Ergebnisse durch numerische Experimente verifiziert.

Abstract

Today’s most common circuit models increasingly tend to loose their validity in cir-cuit simulation due to the rapid technological developments, miniaturization and highercomplexity of integrated circuits. This has motivated the idea of combining circuitsimulation directly with distributed device models to refine critical circuit parts.In this thesis we consider a model, which couples partial differential equations for electro-magnetic devices - modeled by Maxwell’s equations -, to differential-algebraic equations,which describe basic circuit elements including memristors and the circuit’s topology.We analyze the coupled system after spatial discretization of Maxwell’s equations ina potential formulation using the finite integration technique, which is often used inpractice. The resulting system is formulated as a differential-algebraic equation with aproperly stated leading term. We present the topological and modeling conditions thatguarantee the tractability index of these differential-algebraic equations to be no greaterthan two. It shows that the tractability index depends on the chosen gauge conditionfor Maxwell’s equations.For successful numerical integration of differential-algebraic equations the index char-acterization plays a crucial role. The index can be seen as a measure of the equation’ssensitivity to perturbations of the input functions and numerical difficulties such as thecomputation of consistent initial values for time integration.We generalize index reduction techniques for a general class of differential-algebraicequations by using the tractability index concept. Utilizing the index reduction we de-duce local solvability and perturbation results for differential-algebraic equations havingtractability index-2 and we derive an algorithm to calculate consistent initializations forthe spatial discretized coupled system.Finally, we demonstrate our results by numerical experiments.

Acknowledgement

This thesis was written during my employment at the Chair of Mathematics/Numericsat the University of Cologne.

First of all special thanks are due to Prof. Dr. Caren Tischendorf for all the interestingtasks, her guidance, faith and fruitful discussions. It was a great opportunity to havebeen member of her working group.

I also would like to thank Prof. Dr. Ulrich Trottenberg for serving as referee of mythesis, Prof. Dr. Rainer Schrader, who agreed to be the chairperson of my doctoralcommittee, and Dr. Roman Wienands to be a committee member.

I owe gratitude to my colleagues at the University of Cologne, former and present, - inparticular Lennart Jansen, Michael Matthes, Dr. Michael Menrath, Prof. Dr. MonicaSelva Soto and Leonid Torgovitski - for all the discussions on differential-algebraic equa-tions and mathematics in general, but also for being friends with whom to work withhas always been a pleasure.

Also I would like to show my gratitude to the colleagues of the Cologne branch ofthe Fraunhofer-Institute for Algorithms and Scientific Computing SCAI for providing apleasant work environment, in particular Dr. Tanja Clees, Dr. Bernhard Klaassen andClemens-August Thole.

My colleagues in Wuppertal and Darmstadt - Dr. Andreas Bartel, Prof. Dr. MarkusClemens and Prof. Dr. Sebastian Schops - all played an important role in this research.I thank them all for the interesting and encouraging discussions on Maxwell’s equations.

Dr. Wim Schonemaker from MAGWEL owes my thanks for sharing his thoughts andexperience in electromagnetic device simulation with me.

I would also like to thank Yvonne Havertz for proofreading the thesis.

Special thanks are also directed to my parents who have always supported me. I amgrateful to my family and the Glaser family for their outstanding support.Finally, I wish to thank my fiancee Judith Glaser for all her motivational skills, her driveand her patience.

I am indebted to the ICESTARS (FP7/2008/ICT/214911) project, funded by the EU’sSeventh Framework Programme for Research, and the SOFA project (03MS648A), fundedby the German Federal Ministry of Education and Research (BMBF) programme “Math-ematik fur Innovationen in Industrie und Dienstleistungen”.

Frechen, August 27, 2012 Sascha Baumanns

Contents

1 Introduction 1

2 Differential-Algebraic Equations 52.1 Brief Index Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Analysis of Differential-Algebraic Equations . . . . . . . . . . . . . . . . 12

2.2.1 Index Reduction, Solvability and Perturbation Results . . . . . . 202.2.2 Consistent Initialization . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Maxwell’s Equations 493.1 Classical Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.1 Potential Formulation and Gauge Conditions . . . . . . . . . . . . 533.1.2 Boundary and Interface Conditions . . . . . . . . . . . . . . . . . 57

3.2 Finite Integration Technique . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.1 Maxwell’s Grid Equations . . . . . . . . . . . . . . . . . . . . . . 613.2.2 Algebraic Properties of the Discrete Operators . . . . . . . . . . . 673.2.3 Discrete Potential Formulation and Gauge Conditions . . . . . . . 693.2.4 Phantom Objects and Discrete Boundary Conditions . . . . . . . 713.2.5 Maxwell’s Grid Equations with Boundary Excitation . . . . . . . 803.2.6 Numerical Analysis of Maxwell’s Grid Equations . . . . . . . . . . 83

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Electric Network 934.1 Network Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.1.1 Basic Electric Elements . . . . . . . . . . . . . . . . . . . . . . . . 944.1.2 Memristors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.1.3 Network Topology and Kirchhoff Laws . . . . . . . . . . . . . . . 97

4.2 Modified Nodal Analysis for Circuits including Memristors . . . . . . . . 994.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 Coupled Electromagnetic Field/Circuit Models 1135.1 Simulation of Coupled Systems . . . . . . . . . . . . . . . . . . . . . . . 1145.2 Electromagnetic Field/Circuit Model . . . . . . . . . . . . . . . . . . . . 1165.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3.1 Field/Circuit System using Coulomb Gauge . . . . . . . . . . . . 1185.3.2 Field/Circuit System using Lorenz Gauge . . . . . . . . . . . . . 126

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6 Numerical Examples 1356.1 Index Behavior of Field Problems . . . . . . . . . . . . . . . . . . . . . . 1356.2 Memristive Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.3 Coupled Field/Circuit Problems . . . . . . . . . . . . . . . . . . . . . . . 1396.4 Implementation Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Conclusion and Outlook 147

A Linear Algebra 149A.1 Properties of Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 150A.2 Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B Graph Theory 157

C Auxiliary Calculations 161C.1 Topological Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161C.2 Electric Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163C.3 Field/Circuit System using Coulomb Gauge . . . . . . . . . . . . . . . . 166C.4 Field/Circuit System using Lorenz Gauge . . . . . . . . . . . . . . . . . . 170

Notation 175

Bibliography 175

List of Figures

2.1 Solution to the standard and proper formulated η-DAE. . . . . . . . . . . 182.2 Index reduced DAE starting with consistent values. . . . . . . . . . . . . 302.3 Index-2 DAE starting with consistent values. . . . . . . . . . . . . . . . . 302.4 Index reduced DAE starting with inconsistent values. . . . . . . . . . . . 31

3.1 Tonti’s diagram or Maxwell’s house. . . . . . . . . . . . . . . . . . . . . . 543.2 Spatial allocation of a primary cell and a dual cell of the grid doublet. . . 613.3 Allocation of the FIT degrees of freedom on the primary grid. . . . . . . 623.4 Operator mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5 Orientation of the curl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.6 Material properties located on the grid. . . . . . . . . . . . . . . . . . . . 653.7 Primary FIT grid with non-phantom objects. . . . . . . . . . . . . . . . . 71

4.1 Symbols of circuit elements. . . . . . . . . . . . . . . . . . . . . . . . . . 944.2 Symbols of memristor elements. . . . . . . . . . . . . . . . . . . . . . . . 964.3 KCL and KVL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4 Example of a V-loop and an I-cutset. . . . . . . . . . . . . . . . . . . . . 100

6.1 Geometry of the copper bar. . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2 Electric field using Lorenz gauge. . . . . . . . . . . . . . . . . . . . . . . 1366.3 Electric field using Coulomb gauge. . . . . . . . . . . . . . . . . . . . . . 1366.4 Memristor examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.5 HP memristor with Roff “ 16e3 Ω and f “ 5 Hz. . . . . . . . . . . . . . . 1376.6 HP memristor with Roff “ 16e3 Ω and f “ 50 Hz. . . . . . . . . . . . . . 1386.7 HP memristor with Roff “ 36e3 Ω and f “ 5 Hz. . . . . . . . . . . . . . . 1386.8 Memristor with memristance. . . . . . . . . . . . . . . . . . . . . . . . . 1396.9 Two interlocking open copper loops. . . . . . . . . . . . . . . . . . . . . . 1406.10 Current through the open copper loops with f “ 1 Hz. . . . . . . . . . . 1406.11 Two interlocking open copper loops with f “ 1e9 Hz. . . . . . . . . . . . 1416.12 Change in the distribution of the currents. . . . . . . . . . . . . . . . . . 1416.13 More complex circuit with EM device. . . . . . . . . . . . . . . . . . . . 1426.14 Equivalent circuit: Currents through the voltage sources. . . . . . . . . . 1426.15 EM device circuit: Currents through the voltage sources. . . . . . . . . . 143

B.1 An undirected graph and digraph. . . . . . . . . . . . . . . . . . . . . . . 159

List of Tables

3.1 Field quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Differential operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Material properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 Discrete field quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Discrete differential operators. . . . . . . . . . . . . . . . . . . . . . . . . 643.6 Discrete material matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . 663.7 Number of non-phantom objects. . . . . . . . . . . . . . . . . . . . . . . 71

1 Introduction

Insofern sich die Satze der Mathematik auf dieWirklichkeit beziehen, sind sie nicht sicher, undinsofern sie sicher sind, beziehen sie sich nicht aufdie Wirklichkeit.

Albert Einstein, 1879-1955

In various fields such as automotive industry or telecommunication technological progressis mainly driven by a rapid development of integrated circuits. The enormous growth ofperformance is based on a higher complexity and packing density of integrated circuitsas well as decreasing spatial scales and increasing frequencies of electronic devices.The miniaturization of the circuits causes an increasing power density, which in turnmakes it necessary for the prediction of the circuits behavior to take, amongst others,into account heating effects, electromagnetic fields and an accurate switching behaviorof semiconductors.

A common tool to predict the behavior of circuits and to reduce the costs of developmentis circuit simulation. Due to the complexity, which arises from up to millions of circuitelements it is absolutely necessary to keep the model sizes as low as possible. Theconsequences are contradicting demands in circuit simulation: On the one hand thephysical behavior of the circuit needs to be described correctly whereas on the otherhand the computing time must be reasonably small.A well established approach, which tries to fulfill both requirements is the modified nodalanalysis, see [CL75, CDK87, DK84]. The modified nodal analysis models the circuit withbasic elements only, such as capacitors, resistors, inductors, voltage and current sources.Complex elements such as semiconductors or even conductors and their interactions,respectively, are modeled by equivalent circuits consisting of basic elements only. Themodeling of equivalent circuits in an appropriate manner is a challenging task leadingto hundreds of model parameters, see [DF06].

Due to decreasing spatial scales and increasing frequencies the device behavior is alsoinfluenced by the surrounding circuitry, for example, by inductive coupling. It happenswith ever greater frequency that these equivalent circuits are not accurate enough anda refined modeling of a particular device is necessary. Consequently, for complex cir-cuits it is recommended to directly combine circuit simulation with device simulation forparticular devices. However, due to up to millions of circuit elements belonging to onecircuit we are restricted to equivalent circuits for most devices. There is a wide range

1

of modeling levels from linear and nonlinear equations to partial differential equationsdepending on the effects to be described, for example, heating [Bar04, Cul09], semicon-ductor behavior [Tis04, Bod07] and electromagnetic fields [Gun01, Ben06, Sch11].

From the circuit designer point of view not only new manufacturing technologies havea great impact on future integrated circuits but also the development of new circuitelements. Such a new element that most likely will be of huge impact is the memristor.In 1971 Leon Chua postulated the theory of such an element to be existing, but onlyin 2008 the first physical model was released by HP Labs, see [Chu71, SSSW08]. Apartfrom the three basic elements, namely, the capacitor, the resistor and the inductor,already discovered in the 18th and 19th century, the memristor is considered the fourthbasic element. This holds true as the behavior of the memristor cannot be reproducedby any circuit using only the other three basic elements, see [Chu71].

The circuit models including refined devices and memristors lead to systems composed oflinear, nonlinear and ordinary differential equations after applying a method of lines. Fora reliable simulation of these systems we are interested in the perturbation sensitivity.Thus, modeling these systems as differential-algebraic equations with a properly statedleading term is an appropriate approach since for certain classes of such differential-algebraic equations it has been shown that backward differentiation formulas and Runge-Kutta methods are stability preserving, see [HMT03a, HMT03b].

There are several different index concepts to characterize a differential-algebraic equa-tion. All concepts are a measure of the difficulties to be found in the numerical simulationsuch as sensitivity to input perturbations. A direct measure of this sensitivity is the per-turbation index, which takes perturbations of the right hand side into account. Theseperturbations result, for instance, from round-off and Newton method errors. However,the perturbation index in general is difficult to determine. For our investigations wechoose the tractability index concept to determine the differential-algebraic equation’ssensitivity with respect to perturbations.

This thesis is based on three basic issues, namely, differential-algebraic equation theory,structural investigations of circuits including memristors and structural investigationsof circuits including refined electromagnetic devices modeled by Maxwell’s equations.

The first basic issue is the differential-algebraic equation theory, which is the basis for ourlater analysis of the extended circuit models. We familiarize with differential-algebraicequations with a properly stated leading term and the index concept up to index-2 used inthis thesis. For a complete overview on that topic we refer to [LMT13]. The differential-algebraic equation analysis is guided by a generalization of the index reduction techniqueby differentiation of differential-algebraic equations without a properly stated leadingterm to differential-algebraic equations with a properly stated leading term lowering theindex down from two to one. We show that the index reduced differential-algebraicequation has index-1 and a properly stated leading term. Utilizing the index reduction

2

1 Introduction

we deduce local solvability and perturbation index-2 for differential-algebraic equationshaving index-2 from well-known index-1 results given in [LMT13]. One of the difficul-ties in solving differential-algebraic equations numerically is to determine a consistentinitial value to start the integration. To solve this problem we derive an algorithm tocalculate consistent initializations for differential-algebraic equations having index-2 anda properly stated leading term by an approach, which is a generalization of the resultsof [Est00]. All results are derived under sufficient structural conditions.

The second basic issue is the structural investigation of circuits including memristors. Weintroduce the characteristic equations and topology for capacitors, resistors, inductors,voltage and current sources known from literature, [CDK87, DK84], and, in addition, forthe memristor, [Chu71]. For circuits without memristors we introduce the modified nodalanalysis and well-known topological index results of [Tis99, ET00]. For a potential useof the memristor in circuit simulation we need to embed the memristor in actual circuitmodels. The nodal analysis method has already been extended by memristor models.The index of the resulting differential-algebraic equation is investigated in [Ria10]. Inthis thesis we extend the modified nodal analysis by memristor models and the structuralproperties of resulting differential-algebraic equation are presented, which leads to anextension of the topological index results for circuits without memristors.

The third basic issue is the structural investigation of circuits including refined devicesmodeled by Maxwell’s equations. First, we investigate Maxwell’s equations. The par-tial differential equations have been postulated by James Clerk Maxwell in the middleof the 19th century and form the basic of the modern theory of electromagnetics, see[Max64]. We take Maxwell’s equations in a potential formulation into account whichare very popular in a broad filed of applications. Various spatial discretizations havebeen studied, see [BP89, StM05, Cle05, CMSW11], and common spatial discretizationsare the cell method [Ton01], a finite-volume method [MMS01] and variants of the finite-element method [Ned80, Bos98, God10]. In the work presented we opt for the finiteintegration technique introduced in 1977 by Thomas Weiland [Wei77] for spatial dis-cretization. Weiland generalized a finite-difference time-domain-scheme of Kane Yee[Yee66], also known as leap-frog scheme, to solve Maxwell’s equations. The potentialapproach results in a suitable description of Maxwell’s equations and provide a naturallink to the concept of potential differences used in circuit simulation. However, the po-tentials are not uniquely defined and a gauge condition is needed, see [Jac98, Bos01].For the finite integration technique, grad-div formulations based on the Coulomb gaugeare already well known, see [CW02, BCDS11]. In this thesis we introduce a new classof gauge conditions in terms of the finite integration technique motivated by a Lorenzgauge formulation. After spatial discretization we analyze the structural properties ofresulting differential-algebraic equation formulated with a properly stated leading term.It turns out that the index of the differential-algebraic equation depends on the cho-sen gauge condition but without exceeding index-2. To concentrate the link to circuitsimulation a suitable boundary excitation and current formulation is deduced. Nextwe investigate coupled electromagnetic device/circuit models with spatially resolved

3

electromagnetic devices, where the electromagnetic devices are described by Maxwell’sequations in a potential formulation spatially discretized by the finite integration tech-nique. In literature coupled magnetoquasistatic device/circuit models are investigated in[HM76, KMST93, DHW04, DW04] and the index of the resulting differential-algebraicequations is discussed in [Tsu02, Ben06, BBS11, Sch11] using certain conductor modelsand different circuit configurations. These results extend the topological index conditionsfor the modified nodal analysis given in [Tis99, ET00]. Our index analysis for coupledelectromagnetic device/circuit models is not restricted to certain conductor models andwe do not suppose that the magnetoquasistatic assumption holds. We deduce thatthe index of the coupled system depends on the chosen gauge. For the coupled elec-tromagnetic device/circuit model using Lorenz gauge we extend the topological indexconditions for the modified nodal analysis. The Coulomb gauge always results in anindex-2 differential-algebraic equation.

All considered differential-algebraic equations from our application areas have a commonstructure such as a properly stated leading term, constant projectors onto/along certainsubspaces, linear index-2 variables and do not exceed index-2. That is, for all resultingdifferential-algebraic equations we obtain solvability results, perturbation index resultsand we can determine consistent initial values. In particular, we show that the per-turbation index coincides with the tractability index and does not exceed perturbationindex-2.

The first chapter is devoted to the description of the differential-algebraic equationswith a properly stated leading occurring in this thesis. The analysis is guided by thetractability index concept up to index-2. We investigate index reduction, solvability andperturbation results. Methods for computing consistent initializations are derived. Thefollowing chapter introduces the fundamentals of Maxwell’s equations using a potentialformulation and discusses boundary and different gauge conditions. We briefly introducethe finite integration technique for spatial discretization. The structural properties ofthe formulated differential-algebraic equations with incorporated boundary conditionsare discussed and we introduce a new class of gauge conditions formulated for the finiteintegration technique. Index results using different gauge conditions are derived. Chap-ter 3 is devoted to a detailed network analysis. The modified nodal analysis includingmemristor models is derived and new topological index criteria and structural propertiesof the resulting differential-algebraic equations are deduced. In chapter 4 we investigatethe coupled system consisting of circuits refined by spatially resolved electromagneticdevices modeled by the modified nodal analysis and Maxwell’s equations. We generalizethe topological index criteria for the modified nodal analysis for this coupled system andpresent its structural properties. The final chapter provides proof-of-concept examplesto verify the different models including memristors and electromagnetic devices. Ap-pendix A and B subsume the basic aspects of linear algebra and graph theory relevantfor this thesis. Appendix C collects auxiliary calculations needed to prove the indexresults.

4

2 Differential-Algebraic Equations

Differential/Algebraic Equations are not ODE’s.

Linda Petzold, [Pet82]

Two major application fields of differential-algebraic equations are the simulation ofelectric networks and constrained systems. These application areas in engineering canbe seen as important impulse to start with a systematic differential-algebraic equationresearch, since failures in numerical simulations have provoked to analyze these equa-tions.During the last three decades considerable progress in differential-algebraic equationtheory has been made and we refer to [GM86, HLR89, BCP96, ESF98, AP98, HNW02,RR02, KM06, Ria08, LMT13].

Mostly differential-algebraic equations occur because of simplifications of the real prob-lem. In electric networks Kirchhoff current law gives rise to algebraic relationships. Ifthese were modeled either as they are really found or with less idealizations we wouldobtain an ordinary differential equation or a partial differential equation. In mechani-cal systems the simple pendulum model has a fixed constraint on the pendulum lengthwhereas any real material will stretch very slightly, see [Gea06].

Differential-algebraic equations are known to be ill-posed in the sense of Hadamard.This ill-posedness is characterized by the differential-algebraic equation index. Brieflyspeaking, the index can be seen as a measure of the systems’ sensitivity to input per-turbations, as a measure of the difficulties to be found in the numerical simulation andas the difference to an ordinary differential equation. Depending on the point of viewseveral index definitions exist which mostly generalize the Kronecker index in the lineartime-independent case.

In this chapter we introduce the basic notation and tools for the analysis of differential-algebraic equations with a properly stated leading term which occur in this thesis. First,we establish the abstract term of a differential-algebraic equation and point out themain problems we face when dealing with differential-algebraic equations. Second, webriefly introduce some well-known index concepts from literature. Then we familiarizewith differential-algebraic equations having a properly stated leading term guided bythe tractability index concept. Lastly, we deduce new results in index reduction, lo-cal solvability, perturbation results and consistent initialization of differential-algebraicequations.

5

An implicit ordinary differential equation (ODE - Ordinary Differential Equation) is anequation of the form

f

ˆ

d

dty, y, t

˙

“ 0, (2.1)

where f P C pRn ˆD ˆ I,Rnq is given and y P C1 pI,Dq denotes the unknown functionwith D Ă Rn, I Ă R. In this thesis we restrict ourselves to initial value problems, (IVP- Initial-Value Problem) of the form

f

ˆ

d

dty, y, t

˙

“ 0 with y pt0q “ y0 P D and t0 P I.

Definition 2.1. Let be py, tq P D ˆ I with D Ă Rn and I Ă R. We call the implicitODE (2.1) a differential-algebraic equation, (DAE - Differential-Algebraic Equation) iff P C pRn ˆD ˆ I,Rnq, the continuous partial derivatives B

Byf pz, y, tq and B

Bzf pz, y, tq

exist and, in addtion, the partial derivative B

Bzf pz, y, tq is singular with constant rank for

all pz, y, tq P Rn ˆD ˆ I.

DAEs have, amongst others, the following two important properties:

Several components of the solution are determined by constraints. For IVPs theseconstraints limit the choice of initial values since there is not a solution throughevery given initial value.

DAEs with an index higher than two do not only represent integration problemsbut also differentiation problems. This implies that some parts of the DAE mustbe differentiable sufficiently often and the differentiations and integrations may beintertwined in a complex manner.

The behavior of DAEs differs from that of explicit ODEs in several aspects. In thefollowing we describe some of the essential differences.

Example 2.2. Regarding the DAE

d

dty2 “ y2 ` y1 y1 “ q ptq

with the solution

y1 “ q ptqand y2 being the solution to the explicit ODE d

dty2 “ y2`q ptq for a given input function

q. The solution has the properties:

Only y2 has to be continuously differentiable with respect to t.

The initial value for y1 is fixed by the input function q.

6


Example 2.3. Regarding the DAE

d

dty1 “ y2 ´ y1 y1 “ q ptq

with the solution

y2 “ d

dtq ptq ` q ptq y1 “ q ptq

where q is a given input function. The solution has the properties:

The input function q has to be continuously differentiable with respect to t.

The initial values are completely fixed by the input function q and ddt

q.

To get a solution to y2 we need to differentiate y1 with respect to t.

Example 2.4. DAEs are are ill-posed problems. Regarding

d

dtx1 “ x2 and x1 “ sinptq ` δptq

where δptq is a perturbation of the system, the solution is given by

x1 “ sinptq ` δptq and x2 “ cosptq ` d

dtδptq.

A very small perturbation δptq, for example δptq “ 10´k sinp102ktq with k " 1, canhave a serious impact on the solution when compared to the solution x2 “ cosptq of theunperturbed problem, where δ “ 0, since d

dtδptq “ 10k cosp102ktq.

2.1 Brief Index Survey

. . . and please no war between the different indexcamps . . .

Andreas Griewank during the “TwelfthEuropean Workshop on AutomaticDifferentiation with Emphasis on

Applications to DAEs”, Dec. 09, 2011,Berlin.

In this section, we briefly introduce some well-known index concepts from literaturenamely the Kronecker index, the differential index, the perturbation index and thestrangeness index. In the majority of cases DAEs arising in industrial applicationsare nonlinear. The Kronecker index is only defined for linear DAEs with constant co-efficients. Other index definitions are mostly generalizations of the Kronecker index forthe time-varying and nonlinear case. The different index definitions depend on variousperspectives but all concepts exist in their own right and each has its own pros and cons.

7

Kronecker Index

The first introduced index concept was the Kronecker index [GP83, GM86]. The conceptis only defined for linear DAEs (2.1) with constant coefficients given by

Ad

dty ` By “ q (2.2)

with A,B P Rnˆn, q P C pI,Rnq and y P C1 pI,Dq, where A is singular. For this type ofDAEs the Kronecker index provides a closed solution formula. The Kronecker index isclosely related to regular matrix pencils.

Definition 2.5 ([Gan71]). Let be A,B P Rnˆn. The ordered matrix pair tA,Bu and thematrix pencil λA` B respectively are called nonsingular if there is a constant λ P R sothat det pλA` Bq ı 0. Otherwise they are called singular.

Both the ordered matrix pair tA,Bu and the linear DAE (2.2) are said to be regularif the accompanying matrix pencil is nonsingular. In fact, the regular matrix pencil isessential for the unique solvability of the DAE (2.2), see [LMT13].

Lemma 2.6. If the matrix pair tA,Bu is nonsingular, then 1hA ` B is nonsingular for

sufficiently small h ą 0.

Proof . We regard the polynomial det pλA` Bq in λ. If det pλA` Bq ı 0 then thereis only a finite number of roots of the polynomial. Let λ0 be the root with the largestabsolute value. Then 1

hA` B is nonsingular for all 0 ă h ă 1

|λ0|.

Theorem 2.7. For any regular matrix pair tA,Bu there are nonsingular matrices L,K PRnˆn and an integer 0 ď l ď n such that

LAK “„

I 00 N

and LBK “„

W 00 I

(2.3)

with N P Rlˆl and W P Rpn´lqˆpn´lq. Here, N is absent if l “ 0. Otherwise thereis 0 ď k ď l such that N is nilpotent of order k, that is, Nk “ 0 and Nk´1 ‰ 0.The integers l and k as well as the eigenstructure of the blocks N and W are uniquelydetermined by the matrix pair tA,Bu.Proof . See Proposition 1.3 in [LMT13].

The matrix N in Theorem 2.7 has only the eigenvalue zero and can be transformed intoits Jordan normal form by means of a real valued similarity transformation. Therefore,the transformation matrices L and K can be chosen such that N is in Jordan form. Thepair given by (2.3) is called Weistraß-Kronecker form of the regular matrix pair tA,Bu,see [Gan71].

Definition 2.8 (Kronecker index ). The Kronecker index of a matrix regular pair tA,Buand the Kronecker index of a regular DAE (2.2) are defined to be the nilpotency orderk in the Weistraß-Kronecker form (2.3).

8


The DAE (2.2) in Weistraß-Kronecker form is completely decoupled and provides a broadinsight into the structure of the DAE. Every regular DAE (2.2) with Kronecker index-kcan be transformed into the Weistraß-Kronecker form given by

d

dtu`Wu “ p (2.4)

Nd

dtv ` v “ r (2.5)

with y “ K

ˆ

uv

˙

and Lq “ˆ

pr

˙

. We obtain the explicit ODE (2.4) and for l ą 0 we

deduce from (2.5) the unique solution

v “k´1ÿ

i“0

p´1qi Nirpiq, (2.6)

provided r P Ck´1 pI,Rq by recursive use of (2.5), see [LMT13]. Thus (2.6) shows thedependence of the solution y on derivatives of the right-hand side or the perturbationterm q.The higher the index the more differentiations are needed. From the numerical pointof view it is very important to know the index of a DAE (2.2) as well as details onthe structure responsible for differentiations. The regularity of the matrix pair tA,Buguarantees the unique solvability for linear constant DAE (2.2) if we assume smoothinput functions q. If A and B are time-dependent this unfortunately holds no longertrue. There are examples, where the matrix pair is regular for all t P I and the DAEhas infinitely many solutions. It may also happen that the matrix pair is singular and aunique solution exists, see [BCP96].

If r P Ck pI,Rq, the differentiation of (2.6) yields an ODE for v. That idea is pickedup by the differentiation index that figures out how many differentiations are necessaryto transform the DAE (2.2) into an ODE. On the other hand the perturbation indexdirectly measures the impact of perturbations on the solution.

Differentiation Index

The best known index is probably the differentiation index [Cam87, BCP96]. It is moreor less the number of differentiations needed to transform a DAE into an ODE. Thedifferentiation index received much attention and it is widely used. But it assumes highsmoothness of the DAE which often does not hold for applications.

Definition 2.9 (differentiation index ). The DAE (2.1) has differentiation index-k iff P Ck pRn ˆD ˆ I,Rnq and k is the minimal number of analytical differentiations withrespect to t needed to determine an ODE for d

dty as a continuous function in y and t by

algebraic manipulations only.

9

A major drawback in application of the differentiation index is that the calculated dif-ferentiation index-k is just an upper bound for the exact differentiation index of thesystem and the exact index can be lower than k. The calculated differentiation indexdepends strongly on a successful rearranging of the system’s unknowns. However, thedifferentiation index is not clearly defined for DAEs as

yd

dty “ 0.

Here y ” 0 is a solution, but then we can not determine an ODE for y. Otherwise fory ” 1 we obtain d

dty “ 0 and hence differentiation index-1.

Perturbation Index

The perturbation index [HLR89, HNW02] interprets the index as a measure of sensitivityof the solution with respect to perturbations of the given problem and the right handside. The perturbation may arise from rounding errors and numerical approximations.From a numerical point of view the perturbation index is the most important one. Amajor drawback is that the perturbation index does not give us a prescription way howto determine it and requires knowledge about the exact solution.

Definition 2.10 (perturbation index ). The DAE (2.1) has perturbation index-k alonga solution y˚ P C1 pI,Dq on a compact interval I “ rt0, T s, if k is the smallest numberso that for all functions y P C1 pI,Dq with

f

ˆ

d

dty, y, t

˙

“ q ptq

for q P Ck´1 pI,Rnq and all t P I, the inequality

y ´ y˚8 ď c

˜

y pt0q ´ y˚ pt0q8 `k´1ÿ

j“0

›

›qpjq›

›

8

¸

holds true for some c ą 0 as long as›

›qpjq›

›

8, j ă k, and y pt0q ´ y˚ pt0q8 are sufficiently

small.

Strangeness Index

The strangeness index [KM94, KM06] is an algorithmic approach, relies on a transforma-tion to a canonical form and is closely related to the differentiation index, but extendedto over- and under-determined systems.We will briefly review some of the key properties related to the strangeness index, see[Voi06]. From the Kronecker index point of view two matrix pairs tA1,B1u and tA2,B2uare considered to be equivalent if there exist nonsingular matrices U and V, so that

A2 “ UA1V and B2 “ UB1V.

10


For A1 “ A and B1 “ B the DAE (2.2) is rewritten in terms of the transformed unknownx “ V´1y. If V depends on the time, then y “ Vx needs to be differentiated to obtainthe transformed DAE. Thus, d

dty “ d

dtVx ` V d

dtx holds true and the additional term

ddt

Vx has to be taken into account.The strangeness index concept considers two time-dependent matrix pairs tA1,B1u andtA2,B2u, with Ai,Bi P C pR,Rnˆmq and i “ 1, 2, equivalent, if there exist point-wisenonsingular matrix functions U P C pR,Rnˆnq and V P C pR,Rmˆmq, so that

A2 “ UA1V and B2 “ UB1V ` UA1d

dtV.

Under certain constant rank assumptions it is possible to derive the normal form

tA1,B1u „ tA2,B2u “

$

’

’

’

’

&

’

’

’

’

%

»

—

—

—

—

–

Is 0 0 00 Id 0 00 0 0 00 0 0 00 0 0 0

fi

ffi

ffi

ffi

ffi

fl

,

»

—

—

—

—

–

0 A12 0 A14

0 0 0 A24

0 0 Ia 0Is 0 0 00 0 0 0

fi

ffi

ffi

ffi

ffi

fl

,

/

/

/

/

.

/

/

/

/

-

s1

d1

a1

s1

m1 ´ d1 ´ a1

where the blocks

A12 P C`

R,Rs1ˆd1˘

,A14 P C`

R,Rs1ˆpm´m1´d1´a1q˘

and A24 P C`

R,Rd1ˆpm´m1´d1´a1q˘

are again matrix functions. The numbers s1, d1 and a1 are invariants of the equivalencerelations, see [KM06]. The corresponding linear DAE (2.2) is found to be equivalent tothe DAE:

d

dty1 ` A12y2 ` A14y4 “ q1 (2.7a)

d

dty2 ` A24y4 “ q2 (dynamic part)

y3 “ q3 (algebraic part)

y1 “ q4 (2.7b)

0 “ q5 (consistency condition)

The “strangeness” is derived from the coupling between (2.7a) and (2.7b). Differentiating(2.7b) and inserting into (2.7a) leads to an algebraic equation and we get the DAE:

A12y2 ` A14y4 “ q1 ´ d

dtq4

d

dty2 ` A24y4 “ q2

y3 “ q3

y1 “ q4

0 “ q5

11

The resulting modified matrix pair is again denoted by tA2,B2u. The procedure toobtain a normal from and the elimination of the “strangeness” can be repeated to obtaina sequence of characteristic values psi, di, aiq for the matrix pairs tAi,Biu.A matrix pair tAi,Biu is called strangeness-free if si “ 0. The strangeness index is i,if i P N is the smallest number so that the matrix pair tAi,Biu is strangeness-free, see[KM06].

The strangeness index is a powerful tool for the analysis of DAEs, including over- andunder-determined systems. The resulting normal forms provide much inside into thestructure of a given DAE. But even for simple DAEs it may be difficult to calculate thenormal forms, in particular for nonlinear problems.

Other Index Concepts

Thanks to Caren we have all the numericalproblems.

Andreas Steinbrecher during the“Twelfth European Workshop on

Automatic Differentiation with Emphasison Applications to DAEs” as response to

Caren Tischendorf’s example pointing outa gap in the structural index concept,

Dec. 9, 2011, Berlin.

In addition to the index concepts already mentioned a geometrical theory to studyDAEs as ODEs on manifolds is provided by the geometrical index [RR90, RR02]. Acombinatorial index concept is the structural index [BMR00, Pry01]. In [Jan12a] anew index concept is introduced combining the ideas of the tractability index and thestrangeness index. For more index discussions we refer to [Voi06, Meh12].

Expect the structural index all index concepts mentioned are generalizations of theKronecker index in case of linear DAEs (2.2) with constant coefficients.

2.2 Analysis of Differential-Algebraic Equations

Actually, DAEs ARE ODEs but those whichcannot be solved with respect to x1.

Eberhard Griepentrog, Michael Hankeand Roswitha Marz, [GHM92]

Apart from the index concepts discussed so far there is the tractability index concept.The tractability index is a projector-based algorithmic decoupling concept working in

12


terms of the original unknowns and it is straightforward to determine the tractability in-dex at least in theory. The concept behind the tractability index is a stepwise projectionof the solution onto certain invariant subspaces leading to a precise solution description,see [LMT13]. It focuses on the linearization of a DAE and requires only weak smooth-ness conditions. The decoupling procedure provides a detailed insight into the structureof a given DAE, see [GM86, Ria08, LMT13]. In addition for some classes of DAEs wecan connect the tractability index with the perturbation index, which is of importancefor our purposes.

Quasilinear DAEs (2.1) can be written as

A py, tq d

dty ` b py, tq “ 0,

where A P C pD ˆ I,Rnˆnq and b P C pD ˆ I,Rnq. Formally, it must be assumed that thesolution y P C1 pI,Dq is more smooth than actually required since A py, tq is singular andhence only the solution components in the cokernel of A py, tq have to be continuouslydifferentiable. To avoid the unnecessary smoothness we focus on DAEs (2.1) of the morespecial form

A py, tq d

dtd py, tq ` b py, tq “ 0 (2.8)

with

A P C pD ˆ I,Rnˆmq, d P C1 pI,Rmq, b P C pD ˆ I,Rnq, a properly stated leading term, see Definition 2.11,

and

the continuous partial derivatives B

ByA py, tq, B

Byd py, tq, B

Btd py, tq and B

Byb py, tq exist.

We denote D py, tq “ B

Byd py, tq. The leading term d py, tq figures out precisely which

derivatives are actually involved and we need the continuous differentiability of combi-nations of the solution components only in the cokernel of A py, tq.Definition 2.11 ([Mar01]). The DAE (2.8) has a properly stated leading term if aprojector R P C1 pI, Rmˆmq exists with

ker A py, tq “ ker R ptq and im D py, tq “ im R ptqfor all py, tq P D ˆ I.

Hence ker A py, tq and im D py, tq do not depend on y P D for a DAE with a properlystated leading term and the subspaces have constant dimensions. Furthermore they arewell matched together without any overlap or gap. Once again the leading term showsprecisely all involved derivatives. Due to the historical development most DAEs areformulated without a properly stated leading term.

13

Lemma 2.12. For the DAE (2.8)

ker A py, tq ‘ im D py, tq “ Rm (2.9)

holds true, which is equivalent to

im A py, tq “ im A py, tqD py, tq and ker D py, tq “ ker A py, tqD py, tq .

In addition the identities A py, tqR ptq “ A py, tq and R ptqD py, tq “ D py, tq hold truefor all py, tq P D ˆ I.

Proof . See Lemma A.1.3 in [LMT13] and Lemma A.9.

Definition 2.13. A function y P C pI,Rnq is said to be a solution to the DAE (2.8) ify P C1

d pI,Dq with the canonical solution set

C1d pI,Dq “

y P C pI,Dq | d py p¨q , ¨q P C1 pI,Rmq(

and the DAE (2.8) is fulfilled pointwisely.

The solution set of the DAE (2.8) is nonlinear if d py, tq is nonlinear with respect to y.Fortunately it is straightforward to transform the DAE (2.8) into a DAE of the form

A py, tq d

dt

“

D ptq y‰` b py, tq “ 0 (2.10)

with a properly stated leading term as shown in the following. The transformation toa DAE (2.10) makes useful implication such as a solvability and perturbation resultavailable for the DAE (2.8) for a certain DAE class as we will see in the next section.

Definition 2.14. A function y P C pI,Rnq is said to be a solution to the DAE (2.10) ify P C1

DpI,Dq with the canonical linear solution space

C1DpI,Dq “

y P C pI,Dq | D p¨q y p¨q P C1 pI,Rmq(

and the DAE (2.10) is fulfilled pointwisely.

This allows linearization of the DAE which based on linear function spaces, see [Mar01].In fact, C1

DpI,Dq is a vector space over R using point-wise addition and scalar multipli-

cation. Together with the norm

yC1D

“ y8`›

›

›

›

d

dt

“

D ptq y‰

›

›

›

›

8

we obtain the Banach space´

C1D, ¨C1

D

¯

, see Theorem 9 in [GM86].

14


Definition 2.15. The natural extension of a DAE (2.8) is given by

A py, tq d

dt

“

D ptq y‰` b py, tq “ 0 (2.11)

with

y “„

yz

, A py, tq “„

A py, tq0

, D ptq “ “

0 R ptq‰ and b py, tq “„

b py, tqz´ d py, tq

.

The original DAE (2.8) is called the underlying DAE to the natural extension.

The natural extension and the underlying DAE are closely related as shown in the nexttheorem.

Theorem 2.16 ([Mar01]). The natural extension (2.11)

(i) is a DAE of the form (2.10)

(ii) and the underlying DAE are equivalent by the relation z “ d py, tq, t P I.

Proof . (i) Clearly ker A py, tq “ ker A py, tq and im D py, tq “ im D ptq due to the prop-erly stated leading term of the underlying DAE (2.8). Hence we can choose R ptq “ R ptqand the natural extension (2.11) has a properly stated leading term as well.(ii) If y˚ P C1

d pI,Dq is a solution to the DAE (2.8) then y˚ P C1DpI,D ˆ Rnq with

z “ d py, tq, t P I, is a solution to the natural extension (2.11). If y˚ P C1DpI,D ˆ Rnq is

a solution to the natural extension (2.11), then d py, tq “ R ptq z P C1 pI,Rmq holds trueand hence y˚ P C1

d pI,Dq is a solution to the underlying DAE (2.8).

Obviously if y P C pI,Rnq is a solution to the DAE (2.8) then y ptq PM0 ptq for all t P Imust hold true with

M0 ptq “ ty ptq P D | b py, tq P im A py, tqu Ă Rn

is the so-called obvious constraint set, see [Mar03]. The flow of the DAE (2.8) is restrictedto M0 ptq and there is no solution through every given initial value. Thus, for thenumerical integration of the DAE (2.8) we need to start the integration using a suitableintial value.

Definition 2.17. A value y0 P M0 pt0q is said to be a consistent initial value of theDAE (2.8) if there is a solution passing through py0, t0q P D ˆ I.

Definition 2.18. A triple pz0, y0, t0q P RmˆM0 pt0qˆI is said to be an operating pointof the DAE (2.8) if

A py0, t0q z0 ` b py0, t0q “ 0

is fulfilled.

15

The term operating point comes originally from circuit simulation, which is an importantapplication class for this thesis. In some cases, depending on the integration method, weare also interested in a starting value of the derivatives appearing in the DAE (2.8). Thefollowing lemma and definition will characterize the values of the derivatives properly.

Lemma 2.19. For every y0 PM0 pt0q, t0 P I, of the DAE (2.8) there is a unique z0 P Rm

such that

A py0, t0q z0 ` b py0, t0q “ 0 and z0 “ R pt0q z0 (2.12)

are fulfilled.

Proof . Let y0 PM0 pt0q, t0 P I and z10, z2

0 P Rm be fulfilling (2.12). Then

A py0, t0q`

z10 ´ z2

0

˘ “ 0 and z10 ´ z2

0 P ker R pt0q .Furthermore z1

0 ´ z20 “ R pt0q pz1

0 ´ z20q and z1

0 ´ z20 P im R pt0q are valid. Hence we have

z10 ´ z2

0 “ 0.

Definition 2.20. A triple pz0, y0, t0q P im R pt0q ˆM0 pt0q ˆ I is said to be a consistentinitialization of the DAE (2.8) if y0 PM0 pt0q is a consistent value and the triple is anoperating point.

For our investigations we choose the tractability index concept. Once again the tractabil-ity index is a projector-based algorithmic decoupling concept and the concept behind itis a stepwise projection of the solution onto certain invariant subspaces, see [LMT13].In this thesis we focus on DAEs (2.8) of tractability index-1 and index-2 which occurin our applications. Next we define the needed matrices and subspaces for this indexconcept.

Definition 2.21 (Matrix Chain and Subspaces). Given the DAE (2.8) we define:

G0 py, tq “ A py, tqD py, tqB0 pz, y, tq “ B

By rA py, tq z` b py, tqsP0 py, tq “ I´Q0 py, tq , Q0 py, tq projector onto ker G0 py, tqN0 py, tq “ ker G0 py, tq

S0 pz, y, tq “ tv P Rn|B0 pz, y, tq v P im G0 py, tquG1 pz, y, tq “ G0 py, tq ` B0 pz, y, tqQ0 py, tqP1 pz, y, tq “ I´Q1 pz, y, tq , Q1 pz, y, tq projector onto ker G1 pz, y, tqN1 pz, y, tq “ ker G1 pz, y, tqS1 pz, y, tq “ tv P Rn|B0 pz, y, tqP0 py, tq v P im G1 py, tquG2 pz, y, tq “ G1 pz, y, tq ` B0 pz, y, tqP0 py, tqQ1 pz, y, tq

We choose the projector Q1 pz, y, tq such that N0 py, tq Ă ker Q1 pz, y, tq.

16


Remark 2.22. The choice of the projector Q1 pz, y, tq so that N0 py, tq Ă ker Q1 pz, y, tqis always possible. The matrix chain is said to be admissible up to two, see [LMT13].

For computational aspects of the matrix chain as well as for the properly stated leadingterm we refer to [LMT13] taking Remark 2.31 into account.

Definition 2.23 (tractability index ). The DAE (2.8) has (tractability)

index-0 if and only if the index-0 set N0 py, tq satisfies

N0 py, tq “ t0u

index-1 if and only if the DAE (2.8) does not have index-0 and the index-1 setN0 py, tq X S0 pz, y, tq satisfies

N0 py, tq X S0 pz, y, tq “ t0u

index-2 if and only if the DAE (2.8) has neither index-0 nor index-1, the index-1set satisfies

dim pN0 py, tq X S0 pz, y, tqq “ const.

and the index-2 set N1 py, tq X S1 pz, y, tq satisfies

pN1 X S1q pz, y, tq “ t0u

for all pz, y, tq P Rm ˆD ˆ I.

Remark 2.24 ([Mar02]). For the matrix chain of the DAE (2.8) holds:

(i) The projectors Q0 py, tq and Q1 pz, y, tq are not uniquely determined.

(ii) If N0 py, tq Ă ker Q1 pz, y, tq, then Q1 pz, y, tqQ0 py, tq “ 0.

(iii) The index is independent of the choice of the projectors Q0 py, tq and Q1 pz, y, tqas long as (ii) is valid.

Furthermore the index of the DAE (2.8) is invariant under transformation and scaling.

It is worth to formulate a DAE with a properly stated leading term because for a largeclass of index-1 and index-2 DAEs it has been shown that backward differentiationformulas (BDF - Backward Differentiation Formulas) and Runge-Kutta methods arestability preserving, see [HMT03a, HMT03b]. Such a DAE formulation is called numer-ically qualified. An appropriate formulation of the problem ensures a correct behaviorof the numerical solution. The numerical methods keep their stability properties andunexpected step size restrictions can be avoided.

17

Example 2.25. Consider the linear index-1 DAE„

λt λ´ 10 0

d

dt

ˆ

uv

˙

`„

0 0λt´ 1 λ´ 1

ˆ

uv

˙

“ 0 (2.13)

reported [But03] with λ ‰ 1. From the DAE (2.13) we obtain the ODE ddt

u “ λu. Usingthe implicit Euler to solve the DAE (2.13) we obtain

un`1 “ p1` hλq un

which is in fact the explicit Euler applied to the ODE. This will have several consequencessuch as step size restrictions due to stability requirements. Formulating the DAE (2.13)with a properly stated leading term may lead to

„

10

d

dt

ˆ

“

λt λ´ 1‰

ˆ

uv

˙˙

`„ ´λ 0λt´ 1 λ´ 1

ˆ

uv

˙

“ 0. (2.14)

Using the implicit Euler to solve the DAE (2.14) with a properly stated leading term weobtain

un`1 “ 1

1´ hλun

and induce the implicit Euler for the ODE, too.

0.0 0.5 1.0 1.5 2.0 2.5 3.0t

−6

−4

−2

0

2

4

6

v

numerical solutionexact solution

(a) standard formulation

0.0 0.5 1.0 1.5 2.0 2.5 3.0t

0.0

0.2

0.4

0.6

0.8

1.0

v

numerical solutionexact solution

(b) proper formulation

Figure 2.1: BDF-2 solution to the η-DAE with step size h “ 0.01 and η “ ´0.275.

Example 2.26. The well-known linear index-2 DAE, so-called the η-DAE, described in[GP83], further investigated in [HLR89] and successfully tackled by the properly statedleading term in [HMT03b], is given by

„

0 01 ηt

d

dt

ˆ

uv

˙

`„

1 ηt0 1` η

ˆ

uv

˙

“ˆ

e´t

0

˙

18


with the exact solution u ptq “ p1´ ηtq e´t and v ptq “ e´t such that pu0, v0q “ p1, 1qis a consistent initial value at t0 “ 0. Using the original formulation the implicit Eulerfails completely for η “ ´1 and is exponentially unstable for η ă ´1

2, see [GP84]. The

simple reformulation

„

01

d

dt

ˆ

“

1 ηt‰

ˆ

uv

˙˙

`„

1 ηt0 1

ˆ

uv

˙

“ˆ

e´t

0

˙

with a properly stated leading term leads to a correct implicit Euler solution, see[HMT03b]. These statements are confirmed by the numerical results given in Figure 2.1.

Remark 2.27. Let W0 py, tq be a projector along im G0 py, tq. Then

S0 pz, y, tq “ ker W0 py, tqB0 pz, y, tqholds true for all pz, y, tq P Rm ˆD ˆ I.

Remark 2.28. Let W1 pz, y, tq be a projector along im G1 pz, y, tq. Then

S1 pz, y, tq “ ker W1 pz, y, tqB0 pz, y, tqP0 py, tqholds true for all pz, y, tq P Rm ˆD ˆ I.

Now we come back to the relation between the natural extension and their underlyingDAE. The index of the natural extension (2.11) is given by the underlying DAE andvice versa.

Theorem 2.29. The natural extension (2.11) and the underlying DAE have both index-1 or index-2.

Proof . See Theorem 3.4 in [Mar01].

At a later stage we will utilize an equivalent characterization for the tractability index.We use this equivalence to prove that the index reduction introduced in the next sectionreally reduce the index.

Lemma 2.30. The DAE (2.8) has (tractability)

index-0 if and only if G0 py, tq is nonsingular

index-1 if and only if the DAE does not have index-0 and G1 pz, y, tq is nonsingular

index-2 if and only if the DAE has neither index-0 nor index-1 and G2 pz, y, tq isnonsingular

for all pz, y, tq P Rm ˆD ˆ I with constant rank.

Proof . See [GM86] and [Est00].

19

Remark 2.31. Notice that we define G2 pz, y, tq different to [LMT13]. However, since

G2 pz, y, tq “ G2 pz, y, tq pI´ P1 pz, y, tqE pz, y, tqQ1 pz, y, tqqholds true with pI´ P1 pz, y, tqE pz, y, tqQ1 pz, y, tqq nonsingular, see Lemma A.5, it issufficient to check whether G2 pz, y, tq is nonsingular or not in the index-2 case. ForG2 pz, y, tq we have E pz, y, tq “ D py, tq´ d

dt

`

D py, tqP0 pz, y, tqP1 py, tqD py, tq´˘D py, tq.The next lemma is motivated by [Sch03]. We need the lemma to prove Theorem 2.59.In fact the next lemma provides a decoupling into certain solution components as shownlater.

Lemma 2.32. Let the index-2 DAE (2.8) be given. Then

G2 pz, y, tq´1 G0 py, tq “ P1 pz, y, tqP0 py, tq

and

G2 pz, y, tq´1 B0 pz, y, tq “ G2 pz, y, tq´1 B0 pz, y, tqP0 py, tqP1 pz, y, tq`Q1 pz, y, tq `Q0 py, tq

holds true for all pz, y, tq P Rm ˆD ˆ I.

Proof . The first identity is true since

G0 py, tq “ pG0 py, tq ` B0 pz, y, tqQ0 py, tqqP0 py, tq“ G1 pz, y, tqP0 py, tq“ pG1 pz, y, tq ` B0 pz, y, tqP0 py, tqQ1 pz, y, tqqP1 pz, y, tqP0 py, tq“ G2 pz, y, tqP1 pz, y, tqP0 py, tq

and the second one due to

B0 pz, y, tq “ B0 pz, y, tqP0 py, tqP1 pz, y, tq ` B0 pz, y, tqP0 py, tqQ1 pz, y, tq` B0 pz, y, tqQ0 py, tq

“ B0 pz, y, tqP0 py, tqP1 pz, y, tq `G2 pz, y, tqQ1 pz, y, tq `G2 pz, y, tqQ0 py, tq

is valid.

2.2.1 Index Reduction, Solvability and Perturbation Results

Our goal is to describe all constraint sets for index-2 DAEs with a properly stated leadingterm having a special structure and to derive a solvability and perturbation result. Forthis, a suitable tool is the index reduction. The index reduction may be applied to aDAE to lower the index down from an initially higher index. A well known approach

20


is the differentiation of the DAE or of parts of it. Depending on the DAE structurethis approach may lead to a reduction of the index. Here we follow the techniques in[MR99, Est00, Rod00] for DAEs without a properly stated leading term to reduce theindex of a subclass of index-2 DAE (2.8) with a properly stated leading term. For index-2 DAEs of the form (2.10) an index reduction result can be found in [Mar01, Men11].We have already applied these techniques in [Bau08, BST10] for index-2 DAEs of theform (2.8).

The next lemma shows that the obvious constraint set M0 pt0q describes all constraintsin case of an index-1 DAE (2.8).

Theorem 2.33. Let the DAE (2.8) be of index-1 and t0 P I. Then through eachy0 PM0 pt0q passes exactly one solution to the DAE (2.8).

Proof . See Theorem 2.3 in [HM04], using the relation between the DAE (2.8) and itsnatural extension given in Theorem 2.16 and 2.29.

Note that Theorem 2.33 ensures local unique solvability only. For a a global unique solv-ability result for DAEs using the concept of strong monotonicity under certain structuralconditions we refer to [JMT12].

In contrast to index-1 DAEs the flow of the index-2 DAE (2.8) is additionally restrictedby a set M1 ptq, where the relation M1 ptq Ă M0 ptq holds true. For every solutiony P C pI,Rnq of the DAE (2.8) the relation y ptq PM1 ptq, is fulfilled for all t P I. Theset M1 ptq is the index-2 constraint set with M1 ptq “ M0 ptq X H1 ptq, t P I, whereH1 ptq is the so-called hidden constraint set. In case of an index-2 DAE for a consistentvalue y0 “ y pt0q PM1 pt0q holds true for py0, t0q P D ˆ I.

Example 2.34. Consider the index-2 DAE

d

dtu´ u “ 0 (2.15)

vd

dtv ´ vz “ 0 (2.16)

u2 ` v2 ´ 1 “ 0 (2.17)

on D “ tpu, v, zq P R3|v ą 0u. Obviously we get

M0 ptq “ pu, v, zq P R3|u2 ` v2 ´ 1 “ 0

(

.

Differentiation of (2.17) and utilizing (2.15) and (2.16) yields the hidden constraint set

H1 ptq “ pu, v, zq P R3|u2 ` vz “ 0

(

.

That is, the index-2 constraint set is given by

M1 ptq “ pu, v, zq P R3|u2 ` v2 ´ 1 “ 0 and u2 ` vz “ 0

(

.

21

Next we determine the index-2 constraint set M1 ptq by the use of index reduction. LetW0 py, tq and W1 pz, y, tq be projectors along im G0 py, tq and im G1 pz, y, tq, respectively.The projectors W0 py, tq and W1 pz, y, tq will play an important for the index reduction bydifferentiation. At first we need some basic results presented in the next four lemmata.

Lemma 2.35 (Lemma 2.3.1, [Est00]). Let the DAE (2.8) be given. The identities

(i) W1 pz, y, tqW0 py, tq “ W1 pz, y, tq(ii) W1 pz, y, tqB0 pz, y, tqQ0 py, tq “ 0

hold true, for all pz, y, tq P Rm ˆD ˆ I.

Proof . (i) We get

0 “ W1 pz, y, tqG1 pz, y, tqP0 py, tq “ W1 pz, y, tqG0 py, tq .Hence im G0 py, tq Ă ker W1 pz, y, tq that is

ker W0 py, tq “ im G0 py, tq Ă ker W1 pz, y, tqand we conclude

W1 pz, y, tq pI´W0 py, tqq “ 0 ô W1 pz, y, tqW0 py, tq “ W1 pz, y, tq .(ii) We obtain directly

0 “ W1 pz, y, tqG1 pz, y, tq“ W1 pz, y, tq pG0 py, tq ` B0 pz, y, tqQ0 py, tqq“ W1 pz, y, tqB0 pz, y, tqQ0 py, tq .

Lemma 2.36. Let an index-2 DAE (2.8) be given. The identity

im W1 pz, y, tq “ im W1 pz, y, tqB0 pz, y, tqholds true for all pz, y, tq P Rm ˆD ˆ I.

Proof . Clearly im W1 pz, y, tqB0 pz, y, tq Ă im W1 pz, y, tq holds true. From the index-2condition N1 pz, y, tq ‘ S1 pz, y, tq “ Rn can be deduced, see [GM86] using a canonicalprojector

Q1,S pz, y, tq “ Q1 pz, y, tqG2 pz, y, tq´1 B0 pz, y, tqP0 py, tq .Using the Rank–nullity theorem and Lemma 2.35 we can conclude

dim pim W1 pz, y, tqB0 pz, y, tqq “ dim pim W1 pz, y, tqB0 pz, y, tqP0 py, tqq“ n´ dim pker W1 pz, y, tqB0 pz, y, tqP0 py, tqq

22


“ n´ dimS1 pz, y, tq“ dimN1 pz, y, tq“ dim pker G1 pz, y, tqq“ n´ dim pim G1 pz, y, tqq“ n´ dim pker W1 pz, y, tqq“ dim pim W1 pz, y, tqq

and therefore it follows the missing inclusion, see Remark 2.28.

For the next investigations we denote by D py, tq´ a pseudoinverse of D py, tq. To obtaina unique D py, tq´ we choose

D py, tq´ D py, tq “ P0 py, tq and D py, tqD py, tq´ “ R ptqfor all py, tq P D ˆ I, see Theorem A.14 and Lemma A.15.

Lemma 2.37. Let an index-2 DAE (2.8) be given. The identity

im W1 pz, y, tq “ im W1 pz, y, tqB0 pz, y, tqD py, tq´

holds true for all pz, y, tq P Rm ˆD ˆ I with P0 py, tq “ D py, tq´ D py, tq.Proof . Clearly one inclusion is obvious. Let be x P im W1 pz, y, tq. Then there arev, u P Rn and w P Rm such that

x “ W1 pz, y, tq v

“ W1 pz, y, tqB0 pz, y, tqP0 py, tq u

“ W1 pz, y, tqB0 pz, y, tqD py, tq´ D py, tq u

“ W1 pz, y, tqB0 pz, y, tqD py, tq´ w

and hence x P im W1 pz, y, tqB0 pz, y, tqD py, tq´, see Lemma 2.35 and 2.36.

In the following we need d py, tq depends only on dynamic components. This structurewill be exploited later on.

Lemma 2.38. Let the DAE (2.8) be given with P0 P C pI,Rnˆnq and domain D so thatfor each py, tq P D ˆ I also P0 ptq y` sQ0 ptq y P D for all s P r0, 1s. Then, the identities

(i) d py, tq “ d pP0 ptq y, tq(ii) D py, tq “ D pP0 ptq y, tq(iii) B

Btd py, tq “ D pP0 ptq y, tq d

dtP0 ptq y ` B

Btd pP0 ptq y, tq

hold true for all py, tq P D ˆ I.

23

Proof . We use ker P0 ptq “ im Q0 ptq “ ker D py, tq for all py, tq P D ˆ I. (i) We applythe mean value theorem, see [Mar03]. We get

d py, tq ´ d pP0 ptq y, tq “ż 1

0

D psy ` p1´ sqP0 ptq y, tqQ0 ptq yds “ 0.

(ii) We directly obtain

D py, tq “ BByd py, tq “ B

Byd pP0 ptq y, tq “ D pP0 ptq y, tqP0 ptq “ D pP0 ptq y, tq .

(iii) On the one hand we have

d

dtd py, tq “ D py, tq d

dty ` B

Btd py, tq

and on the other hand

d

dtd pP0 ptq y, tq “ D pP0 ptq y, tqP0 ptq d

dty `D pP0 ptq y, tq d

dtP0 ptq y ` B

Btd pP0 ptq y, tq

“ D pP0 ptq y, tq d

dty `D pP0 ptq y, tq d

dtP0 ptq y ` B

Btd pP0 ptq y, tq .

Combining both via ddt

d py, tq “ ddt

d pP0 ptq y, tq we achieve

BBtd py, tq “ D pP0 ptq v, tq d

dtP0 ptq y ` B

Btd pP0 ptq y, tq .

Now we collect all ingredients for the index reduction of index-2 DAEs. The applicationclasses we investigate the forthcoming chapters have special structures. We will restrictourselves to index-2 DAEs (2.8) of the form

A py, tq d

dtd py, tq ` b py, tq “ 0, (2.18)

where we assume

constant projectors Q0 and W1,

domain D so that for each y P D also P0y ` sQ0y P D for all s P r0, 1s, the continuous partial derivative B

BtW1b py, tq exists for all py, tq P D ˆ I

and

by D py, tq´ we denote the pseudoinverse of D py, tq with D py, tq´ D py, tq “ P0 andD py, tqD py, tq´ “ R ptq, see above.

24


Remark 2.39. Since W1 is constant the relation

W1B0 pz, y, tq “ W1BBy rA py, tq z` b py, tqs “ W1

BByb py, tq

holds true.

To extract suitable parts of the DAE (2.18) to reduce the index by differentiation weleft-multiplying the DAE (2.18) by W1 and obtain

W1b py, tq “ 0

due to Lemma 2.35 and ker W0 py, tq “ im G0 py, tq “ im A py, tq. Hence the relation

d

dtW1b py, tq “ 0 (2.19)

holds true, too. The next step is to describe the derivative (2.19) in a proper way so thatthe DAE (2.18) can be reformulated as an index-1 DAE with a properly stated leadingterm.

Lemma 2.40. Let the DAE (2.18) be given. The relation

d

dtW1b py, tq “ W1

BByb py, tqD py, tq´

ˆ

d

dtd py, tq ´ B

Btd py, tq˙

` BBtW1b py, tq

holds true for all py, tq P D ˆ I.

Proof . We apply Lemma 2.35 and 2.38. Since Q0 is constant, it holds that

d

dtd py, tq “ D py, tq d

dtrP0ys ` B

Btd py, tq (2.20)

and

d

dtW1b py, tq “ W1

BByb py, tq d

dtrP0ys ` B

BtW1b py, tq . (2.21)

Left-multiplying of (2.20) by D py, tq´ leads to

d

dtrP0ys “ D py, tq´ d

dtd py, tq ´D py, tq´ BBtd py, tq

since D py, tq´ D py, tq “ P0. Substitution into (2.21) yields the result.

The DAE (2.18) can be written as

0 “ A py, tq d

dtd py, tq ` b py, tq

“ A py, tq d

dtd py, tq ` pI´W1q b py, tq `W1b py, tq

25

and replacing W1b py, tq by ddt

W1b py, tq leads to

A py, tq d

dtd py, tq ` pI´W1q b py, tq ` d

dtW1b py, tq “ 0.

Using Lemma 2.40 we obtain from the index-2 DAE (2.18) the index-1 DAE

A py, tq d

dtd py, tq ` b py, tq “ 0 (2.22)

with

A py, tq “ A py, tq `W1BByb py, tqD py, tq´ ,

b py, tq “ pI´W1q b py, tq ´W1BByb py, tqD py, tq´ BBtd py, tq ` B

BtW1b py, tq

and a properly stated leading term. Within the next lemmata we prove the properlystated leading term and that the DAE (2.22) has indeed index-1.

Lemma 2.41. The DAE (2.22) has a properly stated leading term utilizing the projectorR P C1 pI,Rnˆnq of the DAE (2.18).

Proof . It is sufficient to show the relation ker A py, tq “ ker A py, tq. The first inclusionker A py, tq Ă ker A py, tq follows immediately using the identity

W1BByb py, tqD py, tq´ “ W1

BByb py, tqD py, tq´ R ptq

and ker A py, tq “ ker R ptq. Let be v P ker A py, tq. Using the constant projector W1,see Lemma 2.35, we achieve v P ker W1

B

Byb py, tqD py, tq´ and hence v P ker A py, tq.

That means the decomposition in ker A py, tq and im D py, tq can be realized using theprojector R ptq.The relation ker G0 py, tq “ ker G0 py, tq holds true since the DAEs (2.18) and (2.22) havethe same leading term d py, tq, see Lemma 2.12. That is, the same derivatives occur inboth DAEs.

First, we need a technical lemma to handle second order derivatives with respect to theunknowns to prove the index-1 result for the DAE (2.22).

Lemma 2.42. Let the DAE (2.18) be given. Then, the relations

(i) W1b py, tq “ W1b pP0y, tq(ii) B

By

”

W1B

Byb py, tq z

ı

Q0 “ 0 for all z P Rn


26


Proof . (i) We apply the mean value theorem. We get

W1b py, tq ´W1b pP0y, tq “ż 1

0

W1BByb psy ` p1´ sqP0y, tqQ0yds

“ż 1

0

W1B0 pz, sy ` p1´ sqP0y, tqQ0yds

“ 0.

(ii) We define H pyq “ W1B

Byb py, tq z for fixed z P Rn. Regarding the directional deriva-

tive of H pyq along Q0w for all w P Rn. Using (i) we get

BByH pyqQ0w “ lim

hÑ0

H py ` hQ0wq ´ H pyqh

“ limhÑ0

1

h

„

W1BByb py ` hQ0w, tq z´W1

BByb py, tq z

“ limhÑ0

1

h

BBy rW1b pP0y, tq z´W1b pP0y, tq zs

“ 0

for all w P Rn.

Lemma 2.43. The DAE (2.22) has index-1.

Proof . We compute G1 pz, y, tq by

G1 pz, y, tq “ G0 py, tq ` B0 pz, y, tqQ0

“ G0 py, tq ` BBy

“

A py, tq z` b py, tq‰Q0

“ G1 pz, y, tq `W1BByb py, tqP0

` BBy

„

W1BByb py, tqD py, tq´

ˆ

z´ BBtd py, tq

˙

Q0

deploying Lemma 2.35. Using Lemma 2.42 we get

G1 pz, y, tq “ G1 pz, y, tq `W1BByb py, tqP0.

Finally we have

v P ker G1 pz, y, tq ô v P ker G1 pz, y, tq and v P ker W1B0 pz, y, tqP0

ô v P N1 pz, y, tq X S1 pz, y, tqby using Remark 2.39 and hence v “ 0, because N1 pz, y, tq X S1 pz, y, tq “ t0u due tothe DAE (2.18) having index-2. The decomposition of G1 pz, y, tq can be realized by W1

and we conclude that G1 pz, y, tq is nonsingular, that is, the DAE (2.22) has index-1.

27

To make use of the index reduced DAE (2.22) we need to relate the solution to (2.18)and (2.22). An analytical solution to the index reduced DAE is not necessarily a solutionto the index-2 DAE but the other way holds true. The index reduced DAE has moresolutions than the index-2 one. But if the initial conditions are chosen properly a solutionto the index reduced DAE is a solution to the index-2 DAE, too.

Theorem 2.44. Let W1b py0, t0q “ 0 be fulfilled in one point py0, t0q P DˆI. A solutionto the DAE (2.18) through py0, t0q is a solution to the DAE (2.22) through py0, t0q andvice versa.

Proof . Let y˚ P C1d pI,Dq be a solution to the DAE (2.18). Due to construction the

solution is a solution y˚ to the DAE (2.22), too.The other direction is only true if W1b py0, t0q “ 0 for py0, t0q P D ˆ I and y pt0q “ y0.If the relation d

dtW1b py, tq “ 0 holds true then W1b py, tq “ 0 for all py, tq P D ˆ I. Let

y˚ P C1d pI,Dq be a solution to the DAE (2.22) with y pt0q “ y0. Then

A py˚, tq d

dtd py˚, tq ` b py˚, tq “ 0

and left-multiplication by W1 yields

W1BByb py˚, tqD py˚, tq´

ˆ

d

dtd py˚, tq ´ B

Btd py˚, tq˙

` BBtW1b py˚, tq “ 0

and

d

dtW1b py˚, tq “ 0

respectively, see Lemma 2.40. Due to W1b py0, t0q “ 0 we obtain

W1b py˚, tq “ 0.

Left-multiplying the DAE (2.22) by pI´W1q results in

A py˚, tq d

dtd py˚, tq ` pI´W1q b py˚, tq “ 0.

Hence y˚ is a solution to (2.18).

Now we are able to describe the hidden constraint set H1 ptq and hence the index-2constraint set M1 ptq “M0 ptq XH1 ptq of the DAE (2.18).

Theorem 2.45. The hidden constraint set H1 ptq of the DAE (2.18) can be describedby

H1 ptq “"

y P D|Dz P Rm : W1BByb py, tqD py, tq´

ˆ

z´ BBtd py, tq

˙

` BBtW1b py, tq “ 0

*

.

28


Proof . The set

M0 ptq “

y ptq P D|b py, tq P im A py, tq(

is the obvious constraint set of the index-1 DAE (2.22). Then the index-2 constraint setM1 ptq of the DAE (2.18) can be described by

M1 ptq “

y ptq PM0 ptq |W1b py, tq “ 0(

.

due to Theorem 2.44. Hence

M1 ptq “

y ptq P D|b py, tq P im A py, tq ,W1b py, tq “ 0(

“

y ptq P D|Dz P Rm : A py, tq z` pI´W1q b py, tq ,W1b py, tq “ 0

W1BByb py, tqD py, tq´

ˆ

z´ BBtd py, tq

˙

` BBtW1b py, tq “ 0

(

“M0 ptq XH1 ptqis valid.

Next we deduce an unique solvability result for index-2 DAEs (2.18) using an uniquesolvability result for index-1 DAEs (2.8).

Theorem 2.46. Let a DAE (2.18) be given with t0 P I. Then through each y0 PM1 pt0qpasses exactly one solution.

Proof . Applying the index reduction by differentiation to the DAE (2.18) we get anindex-1 DAE (2.22). Due to Theorem 2.33 for every y0 P M0 ptq we obtain a uniquesolution to the index-1 DAE (2.22). Utilizing Theorem 2.44 leads to the result.

Remark 2.47. The index-2 constraint set M1 ptq of the DAE (2.18) is filled with solu-tions due to the index reduced DAE having index-1 and M1 ptq ĂM0 ptq. Hence everyy PM1 ptq is a consistent value.

As mentioned previously the index reduced DAE has more solutions than the index-2DAE. Hence we need to choose proper initial conditions, see Theorem 2.44. Otherwisewe provoke the so-called drift-off-phenomenon. This is due to differentiating parts of thealgebraic constraints of the index-2 DAE. The former algebraic constraints turns intoODEs for the index reduces DAE. That is, the initial values are not restricted by theformer algebraic constraints anymore and if the initial values violate the former alge-braic constraints then the error increases in time, independent of the step size. Unfortu-nately consistent initial values are not always available. Additionally the used numericalmethod do not necessarily preserve the former algebraic constraints even though theyare preserved in the ODEs with proper initial values. If the step size goes to zero, thedrift-off will go to zero on a fixed time interval, too. To reduce the effect of the drift-offwe can apply projection or stabilizing techniques to correct the algebraic constraintsat certain time points like a Gear-Gupta-Leimkuhler formulation known for multibodysystems, see [Mar96, HNW02, GGL85].

Next we illustrate how the solution to a given example changes due to index reduction.

29

Example 2.48. Consider the index-2 DAE (2.18) given by„

10

d

dt

`“

1 0‰

u˘`

ˆ ´uvu` 1´ f ptq

˙

“ 0

with f ptq ‰ 1 for all t P rt0, T s, T P R. The solution is given by

0.0 0.5 1.0 1.5 2.0 2.5t ×101

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

rel.

erro

r

×10−3

(a) obvious constraints

0.0 0.5 1.0 1.5 2.0 2.5t ×101

−4

−3

−2

−1

0

1

2

3

4

rel.

erro

r

×10−16

(b) hidden constraints

Figure 2.2: Index reduced DAE starting with consistent values.

u ptq “ f ptq ´ 1 and v ptq “ddt

f ptqf ptq ´ 1

with consistent initial values u0 “ f pt0q ´ 1 and v0 “ddt

fpt0q

fpt0q´1. The constraints are given

0.0 0.5 1.0 1.5 2.0 2.5t ×101

−3

−2

−1

0

1

2

3

rel.

erro

r

×10−16


0.0 0.5 1.0 1.5 2.0 2.5t ×101

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

rel.

erro

r

×10−3


Figure 2.3: Index-2 DAE starting with consistent values.

by:

u` 1´ f ptq “ 0 (obvious constraint)

30


uv ´ d

dtf ptq “ 0 (hidden constraint)

Using the constant projectors

0.0 0.5 1.0 1.5 2.0 2.5t ×101

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

rel.

erro

r

×102


0.0 0.5 1.0 1.5 2.0 2.5t ×101

−3

−2

−1

0

1

2

3

rel.

erro

r

×10−16


Figure 2.4: Index reduced DAE starting with inconsistent values.

R “ “

1‰

and P0 “„

1 00 0

we choose

D´ “„

10

and W1 “„

0 00 1

.

This lead to

W1b ppu, vq , tq “ˆ

0u` 1´ f ptq

˙

and

d

dtW1b ppu, vq , tq “

„

01

d

dt

`“

1 0‰

u˘`

ˆ

0´ d

dtf ptq

˙

.

Hence the index reduced DAE reads„

11

d

dt

`“

1 0‰

u˘`

ˆ ´uv´ d

dtf ptq

˙

“ 0

with consistent initial values u0 “ f pt0q ´ 1 and v0 “ddt

fpt0q

fpt0q´1fulfilling the hidden con-

straints. We choose f ptq “ sin ptq ` 3t ` 3, t0 “ 0 and T “ 24. The calculation werecarried out by the implicit Euler scheme using the fixed step size h “ 1e-2. Starting

31

with consistent initial values the difference of the exact and the numerical solution to theconstraints is given in Figure 2.2 and 2.3. To show the drift-off we choose inconsistent

initial values u0 “ ´200 and v0 “ddt

fpt0q

fpt0q´1. The differences in exact and numerical solu-

tions to the constraints are given in Figure 2.4. Note that in case of the index reducedDAE the hidden constraints turn into obvious constraints.

With the index reduction technique we can derive a perturbation result for index-2 DAEs(2.18). First we present a perturbation result for index-1 DAEs (2.8).

Theorem 2.49. Let the DAE (2.8) be of index-1 and I˚ Ă I a compact interval witht0 P I˚. If y˚ P C1

d pI,Dq is a solution to the DAE (2.8), then for all solutions y P C1d pI,Dq

of

A py, tq d

dtd py, tq ` b py, tq “ q ptq ,

the inequality

y ´ y˚8 ď c py pt0q ´ y˚ pt0q8 ` q8qholds true for some c ą 0 as long as q

8and y pt0q ´ y˚ pt0q8 are sufficiently small

and q P C pI˚,Rnq.Proof . See Theorem 4.11 and Remark 4.12 in [LMT13] and using the relation betweenthe DAE (2.8) and its natural extension given in Theorem 2.16 and 2.29.

That is, if the DAE (2.8) has index-1, then the DAE (2.8) has perturbation index-1, seeDefinition 2.10. With that preliminary work we elaborate a new perturbation result forindex-2 DAEs (2.18) using the index reduction techniques.

Theorem 2.50. Let the DAE (2.18) be of index-2 and I˚ Ă I a compact intervalwith t0 P I˚. If y˚ P C1

d pI,Dq is a solution to the DAE (2.18), then for all solutionsy P C1

d pI,Dq of

A py, tq d

dtd py, tq ` b py, tq “ q ptq , (2.23)

the inequality

y ´ y˚8 ď c

ˆ

y pt0q ´ y˚ pt0q8 ` q8 `›

›

›

›

d

dtq

›

›

›

›

8

˙

holds true for some c ą 0 as long as q8

,›

›

ddt

q›

›

8and y pt0q ´ y˚ pt0q8 are sufficiently

small and q P C1 pI˚,Rnq.Proof . Applying the index reduction by differentiation to the DAE (2.18) we get anindex-1 DAE (2.22) with

A py, tq d

dtd py, tq ` b py, tq “ 0 (2.24)

32


and from the perturbed DAE (2.23) we obtain an index-1 DAE (2.22) given by

A py, tq d

dtd py, tq ` b py, tq “ qt (2.25)

with q “ pI´W1q q ` W1ddt

q. Assume q8

,›

›

ddt

q›

›

8and y pt0q ´ y˚ pt0q8 are suffi-

ciently small and q P C1 pI˚,Rnq. Next we apply Theorem 2.49 and we get the theinequality

y ´ y˚8 ď c py pt0q ´ y˚ pt0q8 ` q8q

with c ą 0, where y˚ P C1d pI,Dq is a solution to (2.24) and y P C1

d pI,Dq of (2.25). Thusthe inequality

y ´ y˚8 ď d

ˆ

y pt0q ´ y˚ pt0q8 ` q8 `›

›

›

›

d

dtq

›

›

›

›

8

˙

with d ą 0 holds true, due to the projector W1 being constant.

That is, if the DAE (2.18) has index-2 then the DAE (2.18) has perturbation index-2,too. That is a major justification for choosing the tractability index as index concept.

2.2.2 Consistent Initialization

Image by [HNW02]

An important task for a successful time integration of DAEs is the determination ofa consistent initial value. In this subsection we present methods for the calculation ofconsistent initial values for a subclass of index-1 and index-2 DAEs. For the index-2case we follow the idea of [Est00]. In contrast to [Est00] we elaborate the approach forDAEs with a properly stated leading term.

For the index-1 case a very general approach to calculate consistent initial values can begiven, see [Mar03].

33

Theorem 2.51. Let the index-1 DAE (2.8) be given and t0 P I. The system

A py0, t0q z0 ` b py0, t0q “ 0 (2.26)

pI´ R pt0qq z0 ` R pt0q`

d py0, t0q ´ z0˘ “ 0 (2.27)

is locally uniquely solvable for given z0 P Rm and provides a consistent initializationpz0, y0, t0q P im R pt0q ˆM0 pt0q ˆ I.

Proof . The Jacobian of the nonlinear system (2.26) and (2.26) reads

J pz, yq “„

A py, t0q B0 pz, y, t0qI´ R pt0q R pt0qD py, t0q

.

Let be pz˚, y˚q P ker J pz, yq. Then

A py, t0q z˚ ` B0 pz, y, t0q y˚ “ 0 (2.28)

pI´ R pt0qq z˚ ` R pt0qD py, t0q y˚ “ 0 (2.29)

and we can conclude that y˚ P N0 py, tq X S0 pz, y, tq. That is true since (2.28) leads toB0 pz, y, t0q y˚ P im A py, t0q and (2.29) results in y˚ P ker D py, t0q. We achieve y˚ “ 0because the DAE has index-1. Next (2.28) and (2.29) come to z˚ P im R pt0qXker R pt0q.Hence the Jacobian is nonsingular. From (2.26) we obtain z0 P im R pt0q.In Theorem 2.51 the equation (2.26) ensures that the DAE (2.8) including the obviousconstraints are fulfilled and (2.27) provide the uniqueness of z0.

A difficulty for index-2 DAEs is the description of the so-called index-2 components,which belong to the index-1 set N0 py, tq X S0 pz, y, tq and are determined neither bydifferential nor by derivate-free equations but require inherent differentiation. We call thecomponents of the index-1 set index-2 components since the index-1 set would be empty ifthe DAE may of index-1. To describe the index-2 components we introduce the projectorT pz, y, tq onto N0 py, tq X S0 pz, y, tq and the complementary projector U pz, y, tq “ I ´T pz, y, tq, see [Tis96].

Example 2.52. To clear clear up the misunderstanding that every single solution com-ponent belongs to exactly one index set we inspect the linear index-2 DAE

d

dtx1 ´ x2 ´ x3 “ 0

x1 “ f ptqx2 ´ x3 “ 0

proposed by [Jan12b]. For the description of the index-2 components we make use ofthe relation im Q0Q1 “ N0 X S0, see Lemma 3.5 in [Tis96]. Choosing the projectors

Q0 “»

–

0 0 00 1 00 0 1

fi

fl and Q1 “»

–

1 0 012

0 012

0 0

fi

fl

34


we obtain

N0 X S0 “ span pp0, 1, 1qq “ im T with T “»

–

0 0 00 1 00 1 0

fi

fl ,

that is, the index-2 component is a linear combination of the solution components.

The first relation of the next lemma is taken from [Voi06] and is used to exploit theindex-2 components.

Lemma 2.53. Let a DAE (2.8) be given. The projector U pz, y, tq can be chosen so that

(i) U pz, y, tqP0 py, tq “ P0 py, tq “ P0 py, tqU pz, y, tq(ii) P0 py, tqP1 pz, y, tqU pz, y, tq “ P0 py, tqP1 pz, y, tq

hold true for all pz, y, tq P Rm ˆD ˆ I.

Proof . (i) With im T pz, y, tq Ă ker P0 py, tq we get P0 py, tqT pz, y, tq “ 0. Furthermorewe can choose the projector T pz, y, tq with the property T pz, y, tqP0 py, tq “ 0 due toim P0 py, tq X im T pz, y, tq “ t0u.

(ii) Using (i) we get

P0 py, tqP1 pz, y, tqP0 py, tqU pz, y, tq “ P0 py, tqP1 pz, y, tqP0 py, tq .Furthermore we have

P0 py, tqP1 pz, y, tqP0 py, tq “ pI´Q0 py, tqq pI´Q1 pz, y, tqq pI´Q0 py, tqq“ pI´Q0 py, tqq pI´Q1 pz, y, tqq“ P0 py, tqP1 pz, y, tq

due to Q1 pz, y, tqQ0 py, tq “ 0.

We restrict ourselves to index-2 DAEs (2.18) of the form

A ptq d

dtd py, tq ` b py, tq “ 0, (2.30)

where we assume

N0XS0 py, tq does not depend on py, tq P DˆI and T is a constant projector ontothe index-1 set, U “ I´ T and P0 “ UP0, see [Est00],

domain D so that for each y P D also Uy ` sTy P D for all s P r0, 1s,and

35

index-2 components Ty occur linearly only, that is, the DAE (2.30) can be writtenas

A ptq d

dtd pUy, tq ` b pUy, tq ` B ptqTy “ 0,

where b py, tq “ b pUy, tq ` B ptqTy and B ptq P C pI,Rnˆnq problem given.

In this subsection we develop a step-by-step method to compute consistent initial valuesguided by [Est00]. Note that we extend the approach to DAEs with a properly statedleading term. Under the later structural properties we compute a consistent initializationfor index-2 DAEs as follows:

(i) Describe the hidden constraints.

(ii) Compute an operating point.

(iii) Correct the operating point to fulfill the hidden constraints.

Next we derive the necessary statements for this step-by-step approach. The leadingterm d py, tq of the DAE (2.8) depends only on the non-index-2 components as shownin the following which is an essential ingredient for our investigations. The next twolemmata are used to describe the hidden constraint set H1 ptq given in Theorem 2.45without the index-2 components.

Lemma 2.54. Let the DAE (2.8) be given with P0,U P C pI,Rnˆnq and domain D sothat for each py, tq P Dˆ I also P0 ptq y` sQ0 ptq y P D and U ptq y` sT ptq y P D for alls P r0, 1s. The identities

(i) d py, tq “ d pU ptq y, tq(ii) D py, tq “ D pU ptq y, tq(iii) D py, tq´ “ D pU ptq y, tq´

(iv) B

Btd py, tq “ D pU ptq y, tq d

dtU ptq y ` B

Btd pU ptq y, tq


Proof . Following the proof of Lemma 2.38 and using Lemma 2.53 leads to (i), (ii) and(iv). (iii) Using (ii) the matrix D py, tq´ fulfills the conditions to be a pseudoinverse ofD pU ptq y, tq.Lemma 2.55 (Lemma 2.3.4, [Est00]). Let the DAE (2.8) be given with domain D sothat for each py, tq P DˆI also U ptq y`sT ptq y P D for all s P r0, 1s, W0,U P C pI,Rnˆnqand W1 P Rnˆn. Then

(i) W0 ptqB0 py, tqT ptq “ 0

(ii) W0 ptq b py, tq “ W0 ptq b pU ptq y, tq

36


(iii) W0 ptqB0 py, tq “ W0 ptqB0 pU ptq y, tqU ptq(iv) B

BtW1b py, tq “ B

BtW1b pU ptq y, tq `W1B0 pU ptq y, tq d

dtU ptq y


Proof . (i) We have im T ptq Ă S0 py, tq and ker W0 ptqB0 py, tq “ S0 py, tq, see Re-mark 2.27, that is, W0 ptqB0 py, tqT ptq “ 0.(ii) We apply the mean value theorem. We get

W0 ptq b py, tq ´W0 ptq b pUy, tq “ż 1

0

W0 ptqB0 psy ` p1´ sqU ptq y, tqT ptq yds “ 0

since (i) holds.(iii) We have

W0 ptqB0 py, tq “ BByW0 ptq b py, tq

“ BByW0 ptq b pU ptq y, tq

“ W0 ptqB0 pU ptq y, tqU ptq .(iv) On the one hand we have

d

dtW1b py, tq “ W1B0 py, tq d

dty ` B

BtW1b py, tqand on the other hand applying (iii) we get

d

dtW1b pU ptq y, tq “ W1B0 pU ptq y, tqU ptq d

dty `W1B0 pU ptq y, tq d

dtU ptq y

` BBtW1b pU ptq y, tq

“ W1B0 pU ptq y, tq d

dty `W1B0 pU ptq y, tq d

dtU ptq y

` BBtW1b pU ptq y, tq .

Combining both via ddt

W1b py, tq “ ddt

W1b pU ptq y, tq proves the statement.

We have already described the hidden constraints for the DAE (2.30), see Theorem 2.45,but have not yet taken into account the projector U.

Theorem 2.56. The hidden constraint set H1 ptq of the DAE (2.30) can be describedas

H1 ptq “

y P D|Dz P Rm : W1B0 pUy, tqD pUy, tq´ˆ

z´ BBtd pUy, tq

˙

` BBtW1b pUy, tq “ 0(

.

37

Proof . Let be y P H1 ptq. Then there is z P Rm with

0 “ W1B0 py, tqD py, tq´ˆ

z´ BBtd py, tq

˙

` BBtW1b py, tq ,

see Theorem 2.45. The use of U being constant, W1 “ W1W0 ptq, see Lemma 2.35,

UD pUy, tq´ “ UP0D pUy, tq´ “ P0D pUy, tq´ “ D pUy, tq´ ,due to Lemma 2.53 and P0 “ D pUy, tq´ D pUy, tq, Lemma 2.54 and 2.55 lead to

0 “ W1B0 py, tqD py, tq´ˆ

z´ BBtd py, tq

˙

` BBtW1b py, tq

“ W1B0 pUy, tqD pUy, tq´ˆ

z´ BBtd pUy, tq

˙

` BBtW1b pUy, tq .

Lemma 2.57. Let the DAE (2.30) be given with b py, tq “ b pUy, tq ` B ptqTy. Then

B0 py, tqT “ B ptqT

holds true for all py, tq P D ˆ I.

Proof . We get

B0 py, tq “ BBy

“

b pUy, tq ` B ptqTy‰

“ BBUy

b pUy, tqU` B ptqT.

Thus right-multiplying by T yields B0 py, tqT “ B ptqT, since UT “ 0.

We are ready for the calculation of a consistent initialization in case of an index-2 DAE(2.30). For this we specify and fix all non-index-2 components Uy and try to correctthe index-2 components Ty of the DAE (2.30). At first we need an operating pointpz0, y0, t0q, that is, a triple fulfilling

A pt0q z0 ` b`

Uy0, t0

˘` B pt0qTy0 “ 0, (2.31)

see Definition 2.18. A consistent initialization pz0, y0, t0q, see Definition 2.20, needs tofulfill all constraints and we obtain

A pt0q z0 ` b pUy0, t0q ` B pt0qTy0 “ 0. (2.32)

Due to the fixing of the non index-2 components we have

Uy0 “ Uy0 (2.33)

38


and subtraction of (2.31) from (2.32) using (2.33) yields

A pt0q`

z0 ´ z0˘` B pt0qT

`

y0 ´ y0˘ “ 0. (2.34)

In addition the hidden constraints

W1B0 pUy0, tqD pUy0, t0q´ˆ

z0 ´ BBtd pUy0, t0q

˙

` BBtW1b pUy0, t0q “ 0 (2.35)

are fulfilled, see Theorem 2.56. Using the properties of W1, Lemma 2.35, 2.55, 2.57 and(2.33) the two equations (2.34) and (2.35) are equivalent to

´

A pt0q `W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´¯

z` B pt0qTy “

´W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´

ˆ

z0 ´ BBtd

`

Uy0, t0

˘

˙

´ BBtW1b

`

Uy0, t0

˘

with z “ z0 ´ z0 and y “ y0 ´ y0.

Now we are able to calculate a consistent initialization starting from an operating point.

Theorem 2.58. Let pz0, y0, t0q be an operating point of the index-2 DAE (2.30). Therectangular linear system

´

A pt0q `W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´¯

z` B pt0qTy “

´W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´

ˆ

z0 ´ BBtd

`

Uy0, t0

˘

˙

´ BBtW1b

`

Uy0, t0

˘

Uy “ 0

pI´ R pt0qq z “ ´pI´ R pt0qq z0

has a unique solution pz, yq P Rm`n. A consistent initialization pz0, y0, t0q is given byz0 “ z` z0 and y0 “ y ` y0.

Proof . The proof is divided into three part: Show that the problem has at most onesolution, show that the problem has at least one solution and prove that pz0, y0, t0q is aconsistent initialization.

First we show that the matrix

M “»

–

A pt0q `W1B0 pUy0, t0qD pUy0, t0q´ B pt0qT0 U

pI´ R pt0qq 0

fi

fl (2.36)

is injective, that is,

´

A pt0q `W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´¯

z` B pt0qTy “ 0 (2.37)

39

Uy “ 0 (2.38)

pI´ R pt0qq z “ 0 (2.39)

has only the trivial solution. We split (2.37) into

A pt0q z` B pt0qTy “ 0 (2.40)

W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´z “ 0 (2.41)

using W1, Lemma 2.35 and 2.55. First, we focus on (2.40), which can be rewritten as

0 “ A pt0q z` B pt0qTy

“ A pt0q z` B0

`

y0, t0

˘

Ty

“ A pt0qR pt0q z` B0

`

y0, t0

˘

Ty

“ A pt0qD`

y0, t0

˘

D`

y0, t0

˘´z` B0

`

y0, t0

˘

Ty

“ G0

`

y0, t0

˘

D`

y0, t0

˘´z` B0

`

y0, t0

˘

Ty

by Lemma 2.57. Left multiplying by G2 py0, t0q´1yields

0 “ G2

`

y0, t0

˘´1´

G0

`

y0, t0

˘

D`

y0, t0

˘´z` B0

`

y0, t0

˘

Ty¯

“ P1

`

y0, t0

˘

P0D`

y0, t0

˘´z`G2

`

y0, t0

˘´1B0

`

y0, t0

˘

P0P1

`

y0, t0

˘

Ty

`Q1

`

y0, t0

˘

Ty `Q0Ty,

see Lemma 2.32. Finally, we obtain

0 “ P1

`

y0, t0

˘

D`

y0, t0

˘´z` y, (2.42)

since Lemma 2.53 and (2.38) lead to

Q0T “ T

Ty “ y

Q1

`

y0, t0

˘

T “ 0

P0P1

`

y0, t0

˘

T “ 0

due to Q1 py0, t0qQ0 “ 0. Splitting (2.42) by Q0 and P0 results in

0 “ P0P1

`

y0, t0

˘

D`

y0, t0

˘´z ô 0 “ D

`

y0, t0

˘

P1

`

y0, t0

˘

D`

y0, t0

˘´z (2.43)

and

y “ ´Q0P1

`

y0, t0

˘

D`

y0, t0

˘´z ô y “ Q0Q1

`

y0, t0

˘

D`

y0, t0

˘´z (2.44)

since Q0D py0, t0q´ “ 0 and P0 “ D py0, t0q´ D py0, t0q. By combining (2.39) and (2.43)we gain

z “ D`

y0, t0

˘

Q1

`

y0, t0

˘

D`

y0, t0

˘´z (2.45)

40


because of

z “ R pt0q z

“ D`

y0, t0

˘

D`

y0, t0

˘´z

“ D`

y0, t0

˘

P1

`

y0, t0

˘

D`

y0, t0

˘´z`D

`

y0, t0

˘

Q1

`

y0, t0

˘

D`

y0, t0

˘´z.

Starting from (2.41) using Lemma 2.53, 2.55 and (2.45) we can deduce

0 “ W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´z

“ W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´D`

y0, t0

˘

Q1

`

y0, t0

˘

D`

y0, t0

˘´z

“ W1B0

`

Uy0, t0

˘

P0Q1

`

y0, t0

˘

D`

y0, t0

˘´z

“ W1B0

`

y0, t0

˘

P0Q1

`

y0, t0

˘

D`

y0, t0

˘´z,

that is,

Q1

`

y0, t0

˘

D`

y0, t0

˘´z P ker W1B0

`

y0, t0

˘

P0 “ S1

`

y0, t0

˘

.

Furthermore

Q1

`

y0, t0

˘

D`

y0, t0

˘´z P im Q1

`

y0, t0

˘ “ N1

`

y0, t0

˘

and we conclude Q1 py0, t0qD py0, t0q´ z “ 0 due to N1 py0, t0q X S1 py0, t0q “ t0u sincethe DAE (2.30) has index-2. From (2.44) and (2.45) we end up with pz, yq “ 0. Thatis, the matrix (2.36) is injective and if a solution to the linear system exists then thesolution is unique.

The right-hand side of the linear system reads

g “¨

˝

´W1B0 pUy0, tqD pUy0, t0q´`

z0 ´ B

Btd pUy0, t0q

˘´ B

BtW1b pUy0, t0q

0´pI´ R pt0qq z0

˛

‚.

For the existence of a solution we have to prove g P im M, that is, we have to show thatit exist pz, yq P Rm`n with

»

–

A pt0q `W1B0 pUy0, t0qD pUy0, t0q´ B pt0qT0 U

pI´ R pt0qq 0

fi

fl

ˆ

zy

˙

“ g.

We need y P ker U and z “ v ´ z0 with v P im R pt0q and z0 P Rm. Using W1 we splitthe first equation of the linear system into

A pt0q z` B pt0qTy “ 0 (2.46)

W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´z “ ´W1B0

`

Uy0, t˘

D`

Uy0, t0

˘´ “

z0

41

´ BBtd

`

Uy0, t0

˘ ‰´ BBtW1b

`

Uy0, t0

˘

“ ´W1B0

`

Uy0, t˘

D`

Uy0, t0

˘´ “

R pt0q z0 (2.47)

´ R pt0q BBtd`

Uy0, t0

˘` R pt0qw‰

since D pUy0, t0q´ R pt0q “ D pUy0, t0q´ and it exists w P Rm such that

W1B0

`

Uy0, t˘

D`

Uy0, t0

˘´w “ ´ BBtW1b

`

Uy0, t0

˘

,

see Lemma 2.37. From (2.47) we can deduce

v “ R pt0q BBtd`

Uy0, t0

˘´ R pt0qw P im R pt0qis always a valid choice for fixing the component z. It remains to fix y. For that we haveto investigate (2.46). We get

A pt0q z` B pt0qTy “ A pt0qR pt0q z` B pt0qTy

“ A pt0qR pt0q z` B0

`

y0, t0

˘

Ty

“ G0

`

y0, t0

˘

D`

y0, t0

˘´z` B0

`

y0, t0

˘

Ty

and left-multiplication of G´12 py0, t0q yields

y “ ´TP1

`

y0, t0

˘

D`

y0, t0

˘´z P im T,

see above. We show that g P im M and hence the system has a unique solution.

In the final step we have to show that z0 “ z ` z0 and y0 “ y ` y0 is a consistentinitialization. Since pz0, y0, t0q is an operating point

A pt0q z0 ` b`

Uy0, t0

˘` B pt0qTy0 “ 0

is valid. Addition of A pt0q z` B pt0qTy “ 0 leads to

A pt0q z0 ` b pUy0, tq ` B pt0qTy0 “ 0

due to Uy “ 0, that is, pz0, y0, t0q is an operating point and we get y0 PM0 pt0q. Theequation

0 “ W1B0

`

Uy0, t0

˘

D`

Uy0, t0

˘´

ˆ

`

z` z0˘´ B

Btd`

Uy0, t0

˘

˙

` BBtW1b

`

Uy0, t0

˘

“ W1B0 pUy0, t0qD pUy0, t0q´ˆ

z0 ´ BBtd pUy0, t0q

˙

` BBtW1b pUy0, t0q

ensures that the hidden constraints are fulfilled, that is, y0 PM1 pt0q and pz0, y0, t0q isa consistent initialization since z0 P im R pt0q.

42


The rectangular linear system stated in Theorem 2.58 is well suited for a least squaremethod due to the full column rank.

So far we have not shown how to calculate an operating point pz0, y0, t0q, which is anessential ingredient for Theorem 2.58. Motivated by our application classes we determinean operating point for a subclass of DAEs (2.8) given by

A py, tq d

dtd py, tq ` b py, tq “ 0, (2.48)

where we assume

ker B0 py, tq “ ker B0 py, tqJ independent of y P Rn

and

it exists y0 P Rn such that b py0, t0q “ 0 for t0 P I.

First, we are interested in a point of equilibrium py0, t0q P Rn ˆ I of the DAE (2.48),that is, a point fulfilling b py0, t0q “ 0. If y0 P D, then the point of equilibrium is anoperating point pz0, y0, t0q of the DAE (2.48) with z0 “ 0. For the calculation of a pointof equilibrium for the subclass of DAEs (2.48) we make use of an orthogonal projectordecomposition developed in [Jan12a] for a new index concept.

Theorem 2.59. Let

f py, tq “ 0 with f : Rn ˆ R Ñ Rn

be given. Moreover, let F py, tq “ B

Byf py, tq with ker F py, tq “ ker F py, tqJ be indepen-

dent of y P Rn, BP ptq ““

b1 ptq . . . bk ptq‰ P Rnˆk, where tb1 ptq , . . . , bk ptqu is an

orthonormal basis with respect to the standard scalar product on Rk of ker F py, tq, andBP ptq “

“

bk`1 ptq . . . bn ptq‰ P Rnˆn´k, where tbk`1 ptq , . . . , bn ptqu is an orthonormal

extension of tb1 ptq , . . . , bk ptqu to an orthonormal basis with respect to the standardscalar product on Rn, t P R. In addition we assume the domain D to be so that for eachpy, tq P D ˆ I also PB ptq y ` sQB ptq y P D for all s P r0, 1s with PB ptq “ BP ptqBP ptqJand QB ptq “ I´ PB ptq. Then

BP ptqJ f pBP ptq v, tq “ 0

has a unique solution v P Rk and y “ BP ptq v`u, with u P ker BP ptqBP ptqJ arbitrarily.

Proof . We choose an orthogonal projector PB ptq along ker F py, tq by

PB ptq “ BP ptqBP ptqJ .Using Lemma A.7, A.9, the orthogonality of PB ptq and ker F py, tq “ ker F py, tqJ yieldsim PB ptq “ im F py, tq and

PB ptqF py, tq “ F py, tq “ F py, tqPB ptq

43

for all py, tq P Rn ˆ R. The relation

f py, tq “ f pPB ptq y, tqholds true since applying the mean value theorem provides

f py, tq ´ f pPB ptq y, tq “ż 1

0

F psy ` p1´ sqPB ptq y, tq pI´ PB ptqq yds “ 0.

Due to the choice of BP ptq and BP ptq the matrix C ptq “ “

BP ptq BP ptq‰

is nonsingular.Hence:

f pPB ptq y, tq “ 0 ô C ptqJ f pPB ptq y, tq “ 0 ô#

BP ptqJ f pPB ptq y, tq “ 0

BP ptqJ f pPB ptq y, tq “ 0

Notice that BP ptqJ f pPB ptq y, tq “ 0 holds true for all y P Rn since

BP ptqJ F pPB ptq y, tqPB ptq “ 0

for all py, tq P Rn ˆ I, that is, BP ptqJ f pPB ptq y, tq is independent of y P Rn, and due tothe requirements that y P Rn exists so that f py, tq “ 0 for t P R. Regarding

0 “ BP ptqJ f pPB ptq y, tq“ BP ptqJ f

´

BP ptqBP ptqJ y, t¯

“ BP ptqJ f pBP ptq v, tq

with v “ BP ptqJ y and Jacobian given by

J pvq “ BP ptqJ F pBP ptq v, tqBP ptq .Due to the construction of BP ptq the Jacobian is nonsingular.

Remark 2.60. The orthogonal basis tb1 ptq , . . . , bk ptqu, needed in Theorem 2.59 canbe calculated by, for example, a QR decomposition.

If we are not interested in a consistent initialization of the DAE (2.30) at t0 P I, butfinding a solution satisfying the DAE after the first step, we can also apply the implicitEuler method starting with an operating point y0 P M0 pt0q. Since the DAE (2.30)depends linearly on the index-2 components, the approximation obtained at t1 “ t0 ` his identical to the approximation obtained with a consistent initial value y0 satisfyingUy0 “ Uy0. In detail, if the implicit Euler method is applied to the DAE (2.30) then theapproximation y1 to y pt1q is the solution to the following nonlinear system of equations

f py1q “ 1

hA pt1q pd pUy1, t1q ´ d pUy0, t0qq ` b pUy1, t1q ` B pt1qTy1 (2.49)

44


with f py1q “ 0 and Jacobian J pUy1q “ B

Byf py1q given by

J pUy1q “ 1

hA pt1qD pUy1, t1qU` B

BUyb pUy1, t1qU` B pt1qT.

Applying Newton’s method to (2.49) yields

y11 “ y0

1 ´ J`

Uy01

˘´1f`

y01

˘

and

J`

Uy01

˘

y11 “ J

`

Uy01

˘

y01 ´ f

`

y01

˘

.

That can be reformulated to

J`

Uy01

˘

y11 “ J

`

Uy01

˘

y01 ´ f

`

y01

˘

“ 1

hA pt1qD

`

Uy01, t1

˘

Uy01 ` B

BUyb`

Uy01, t1

˘

Uy01 ` B pt1qTy0

1

´ 1

hA pt1q

`

d`

Uy01, t1

˘´ d pUy0, t0q˘´ b

`

Uy01, t1

˘´ B pt1qTy01

“ 1

hA pt1q

`

D`

Uy01, t1

˘

Uy01 ´ d

`

Uy01, t1

˘` d pUy0, t0q˘

` BBUy

b`

Uy01, t1

˘

Uy01 ´ b

`

Uy01, t1

˘

“ g`

Uy01

˘

and we obtain

y11 “ J

`

Uy01

˘´1g`

Uy01

˘

.

Since this equation depends only on Uy0,Uy01 and Uy0 “ Uy0, the choice y0

1 “ y0

yields the same approximation at t1 as the choice y01 “ y0. The result does not surprise

because it is true for classes of DAEs without a properly stated leading term [Est00].Consequently, py1

1, t1q is a consistent initial value for the DAE (2.30) at t1 P I providedthe rounding errors are zero.

Lemma 2.61. For the DAE (2.30) it is sufficient to start the integration with the implicitEuler using an operating point. All constraints are fulfilled after the first integrationstep.

Remark 2.62. It is a common approach to start the numerical integration with theimplicit Euler to overcome the problem of the calculation of a consistent value, butusually it is not proven that the approach works. Here we have shown that starting theintegration of the DAE (2.30) using the implicit Euler we only need an operating pointand we obtain after one time step a consistent initial value. For this the Theorem 2.58is essential, since here we prove that starting with an operating point only the index-2components Ty has to be correct while the non-index-2 components Uy are fixed.

45

Remark 2.63. The tractability index concept and all techniques presented are notinvariant under transformation with respect to solution components which can be chosenfreely. Consider the index-1 DAE

´x` y “ 0d

dty ` 2x “ 0

where y0 can be chosen arbitrarily within the tractability index concept, but we cannot choose the algebraic component x0. Inserting the first equation into the second oneleads to the index-1 DAE

´x` y “ 0d

dtx` 2x “ 0

where x0 can be chosen freely within the tractability index concept, but we cannot choosethe algebraic component y0. That is dissatisfying since an engineer, for example, doesnot have the immediate possibility to fix certain initial algebraic components such asthe velocity of a car or the energy consumption of an electric device. How to solve thischallenging task is still an open issue and may become subject of future research.

For the index-3 or higher DAEs are more of a challenge. If such a DAE is solved by a BDFmethod, then the solution can have a huge error in the first steps even if consistent initialvalues are given. Instead of consistent initial values we have to introduce numericallyconsistent initial values, that is, values fulfilling the numerical constraints to solve theDAE numerically, see [Are08].

2.3 Summary

In this chapter we have laid the basis of for our later analysis. We have identifiedproblems and challenges in differential-algebraic equation theory and developed andrefined methods to tackle them within the tractability index concept for differential-algebraic equations with a properly stated leading term.The basis was a generalization of the index reduction method by differentiation for index-2 differential-algebraic equations (2.18) to index-1 differential-algebraic equations (2.22)(Lemma 2.41 and 2.43). Next, we have deduced a suitable description of the hiddenconstraints of the index-2 differential-algebraic equations (2.18) (Theorem 2.45), the localunique solvability (Theorem 2.46) and a perturbation index-2 result (Theorem 2.50).The latter is an important justification for the choice of the tractability index conceptin this thesis and is essential to numerics.To start the numerical integration we focused on consistent initial values for differential-algebraic equations (2.30). For index-2 differential-algebraic equations (2.30) we gener-alized a step-by-step approach of [Est00] for differential-algebraic equations without aproperly stated leading term to differential-algebraic equations with a properly statedleading term. For this we had to calculate an operating point. Next the operating pointwas corrected by a full rank linear system providing a consistent initialization (Theo-rem 2.58). For differential-algebraic equations (2.48) we provided a method to compute

46


an operating point (Theorem 2.59). It turned out that for differential-algebraic equa-tions (2.30) it is sufficient to start with an operating point if the implicit Euler methodis used for time integration (Lemma 2.61 and Remark 2.62).

47

3 Maxwell’s Equations

From a long view of the history of mankind, seenfrom, say, ten thousand years from now, there canbe little doubt that the most significant event of the19th century will be judged as Maxwell’s discoveryof the laws of electrodynamics.

The Feynman Lectures on Physics, Volume IIRichard Feynman.

Nowadays electric and magnetic fields are an integral part of our technological life. Weare surrounded by electric and magnetic fields ranging from induction cooking, mobilephones, wireless networks, electric cars to magnetic resonance tomographs.

The reduction of development costs is a core industrial demand. One way to mini-mize efforts is to replace as much laboratory testing as possible by numerical simulationpredicting the full range of device performance. A common device type is the electro-magnetic, which is governed by the interaction between electric and magnetic fields fullyas described by the partial differential equation system of Maxwell’s equations. Forthe numerical simulation of an electromagnetic device we need to discretize Maxwell’sequations in space and time.

A well established method of lines approach for the spatial discretization is the finite inte-gration technique introduced by Thomas Weiland [Wei77] and further developed duringthe last three decades [MW07]. The finite integration technique is used by our partnersin the EU-funded ICESTARS project and the SOFA project, funded by the Germangovernment. Moreover, it is successfully applied in established software packages suchas MAFIA (Technical University Darmstadt) and CST studio (Computer SimulationTechnology AG).

We investigate electromagnetic models described by Maxwell’s equations in a potentialformulation. They are much used in low and high frequency applications with vari-ous formulations and discretizations having already been analyzed, for an overview see[BP89, StM05, Cle05, CMSW11]. Apart from the finite integration technique discretiza-tion the cell method [Ton01], particular finite-volume methods [MMS01] and also certainvariants of the finite-element method are broadly used [Ned80, Bos98, God10]. Here wefocus on the finite integration technique discretization scheme. The potential approachresults in an adequate problem description that provides a natural link to the concept ofpotential differences, which are crucial in circuit simulation. However, the potentials are

49

not uniquely defined and to obtain a consistent description a gauge condition is needed,see [Jac98, Bos01]. For the finite integration technique, grad-div formulations based onthe Coulomb gauge are well understood, see [CW02, BCDS11]. In this thesis we intro-duce a new class of gauge conditions in terms of the finite integration technique drivenby a Lorenz gauge formulation. After spatial discretization we investigate the structuralproperties of the resulting differential-algebraic equation formulated with a properlystated leading term. It turns out that the index of the differential-algebraic equationdepends on the chosen gauge condition but does not exceed index-2. To concentratethe link to circuit simulation a suitable boundary excitation and current formulation isdeduced. Similar differentiation index results are obtained in [BCS12] using a sourceterm excitation and different gauge conditions.

In this chapter the relevant fundamentals of Maxwell’s equations are discussed focusingon the basic features of electromagnetism. First, we analyze the electromagnetic fieldsby using a potential formulation. Different gauge conditions and suitable boundaryconditions are discussed. Second, we briefly introduce the finite integration technique.Especially the structural properties of the discrete operators with incorporated bound-ary conditions are discussed and we introduce a new class of gauge conditions in termsof the finite integration technique. This leads to Maxwell’s grid equations and we derivea current formulation and present a boundary excitation for the potential. Third, theresulting differential-algebraic equations are formulated with a properly stated leadingterm and the new index results are presented, which depend on the chosen gauge con-dition. In addition, we present an approach to calculate a consistent initialization forMaxwell’s grid equations.

3.1 Classical Electromagnetism

Maxwell’s equations (ME - Maxwell’s Equations) are a set of four coupled partial differ-ential equations postulated by James Clerk Maxwell in the middle of the 19th centuryand form the basic of the modern theory of electromagnetics (EM - ElectroMagnetic),see [Max64]. These equations describe all phenomena of EM fields by four vector valuedfunctions of space x P Ω Ă R3 and time t P I Ă R on a simple connected domain Ω. TheEM quantities are denoted by the electric and magnetic field ~E, ~H : Ω ˆ I Ñ R3 andby the electric and magnetic induction ~D, ~B : Ω ˆ I Ñ R3. An EM field is created by,amongst others, a distribution of electric charges and a current flow. The distributionof charges is given by ρ : Ωˆ I Ñ R3 while the conduction current density is describedby ~Jc : Ωˆ I Ñ R3, see [Jac98, HW05].

Today ME in differential form reads:

∇ ¨ ~D “ ρ (3.1)

∇ ¨ ~B “ 0 (3.2)

50


∇ˆ ~E “ ´ BBt~B (3.3)

∇ˆ ~H “ ~Jc ` BBt~D (3.4)

These equations describe the spatial and temporal behavior of the EM quantities. Tablesof SI units are given in Table 3.1 and 3.2.

quantity SI units

~E Vm~D Cm2 “Asm2

~B T“Vsm2

~H Am~Jc, ~Jd, ~Jt Am2

ϕ V~A Wbm“Vsm~Π Vmρ Cm3 “Asm3

Table 3.1: Field quantities.

operator SI unitsB

Bt1s

∇ 1m∇ˆ 1m∇¨ 1m

Table 3.2: Differential operators.

ME are the work of several well-known physicians. That is why the individual equa-tions are attributed to other scientists. But Maxwell grouped all the equations togetherinto a consistent set and introduced the displacement current. The Gauss’ law (3.1)describes the effect of the charge density on the electric induction and Gauss’ law formagnetism (3.2) expresses the fact that magnetic induction is solenoidal. The Maxwell-Faraday’s law (3.3) describes the effect of a time changing magnetic field on the electricfield. Finally, Maxwell-Ampere’s law (3.4) gives the effect of the total current density

on the magnetic field. The total and displacement current density ~Jt,~Jd : Ω ˆ I Ñ R3

are given by

~Jt “ ~Jc ` ~Jd and ~Jd “ BBt~D.

An essential feature of ME is that electric charges are conserved. For this we derivethe charge-current continuity equation from ME. The divergence of (3.4) and the timederivative of (3.1) lead to the continuity equation

∇ ¨ ~Jc ` BBtρ “ 0 (3.5)

expressing the conservation of electric charges. The continuity equation reveals that MEare not independent. If charge is conserved, then Gauss’ law and Gauss’ law for mag-netism are consequences of Maxwell-Faraday’s law and Maxwell-Ampere’s law. Takingthe divergence of (3.3) and (3.4) and interchanging the derivatives we obtain

BBt∇ ¨

~B “ 0 andBBt

´

∇ ¨ ~D´ ρ¯

“ 0

51

using the continuity equation (3.5). Thus, if the divergence conditions (3.1) and (3.2) arefulfilled at one time they hold for all time, see [Mon03]. Hence the divergence conditionsare consequences of the dynamic curl conditions (3.3) and (3.4) and can be seen as re-striction on valid initial conditions for the Maxwell-Faraday’s law and Maxwell-Ampere’slaw. Therefore we conclude that the whole time evolution is completely specified by thedynamic curl conditions. ME are completed by three constitutive laws. The laws relate

quantity SI units

ε Fm=AsVmν mH“AmVsσ Sm“AVm

Table 3.3: Material properties.

~E and ~B to ~D, ~Jc and ~H. These laws depend on the material properties in the domainoccupied by the EM field. One distinguishes between linear and nonlinear, homogeneousand inhomogeneous, isotropic and anisotropic materials. For linear materials the consti-tutive laws are independent of the fields quantities. The constitutive laws of nonlinearmaterials depend on the fields quantities. For homogeneous materials the constitutivelaws are independent on the spatial coordinates. The constitutive laws of inhomogeneousmaterials are functions of the spatial coordinates. Isotropic and anisotropic materialsare characterized by the absence or presence of a dependence of the constitutive lawsupon the spatial direction, see [Ben06].

We restrict ourselves to the following constitutive laws. The first constitutive law relates~E and ~D by

~D “ ε~E, (3.6)

with ε : Ω Ñ R and the permittivity ε depending on the spatial coordinates only.The second constitutive law relates ~B and ~H by

~H “ νp~Bq~B, (3.7)

with ν : Ωˆ R3 Ñ R3ˆ3 and the reluctivity ν depending on the spatial coordinates anddepending nonlinearly and anisotropically on the magnetic induction. The reluctivity isthe inverse of the permeability µ.In case of conductive materials the electric field ~E itself gives rise to a current flow. Thatleads to the last constitutive law also known as Ohm’s law. As long as the field strengthsare not too large we can assume that Ohm’s law is fulfilled. It relates ~E and ~Jc by

~Jc “ σ~E, (3.8)

with σ : Ω Ñ R and the conductivity σ depending on the spatial coordinates only. Ininsulating materials we can assume that σ vanishes.

52


Assumption 3.1 (constitutive laws). The materials have:

(i) Linear, inhomogeneous and isotropic permittivity and conductivity.

(ii) Nonlinear, inhomogeneous and anisotropic reluctivity given by Brauer’s model, see[Sch11, BH91].

Table 3.3 shows the corresponding SI units of the material properties. In this thesis werestrict ourselves to materials fulfilling Assumption 3.1.

Classification of Electromagnetic Problems

The behavior of EM fields is governed by ME. To simplify the calculation of ME thereare several approaches that disregard effects depending on the speed of propagation ofthe EM waves. Common simplifications are:

(i) Static fields: The time dependence in the EM quantities are neglected, that is,B

Bt~B “ 0 and B

Bt~D “ 0.

(ii) Magnetoquasistatic (MQS - MagnetoQuasiStatic): The electric induction ~D is

slowly varying and the time dependence is neglected, that is, B

Bt~D “ 0.

(iii) Electroquasistatic: The magnetic induction ~B is slowly varying and the time de-

pendence is neglected, that is, B

Bt~B “ 0.

Every simplification has an impact on the solution, that is, we have to take care if asimplification is really admissible, see [HM89]. In this thesis we mainly focus on MEwithout simplification, that is, we consider the “full set” of ME in time domain.

3.1.1 Potential Formulation and Gauge Conditions

When studying ME it is often convenient to introduce auxiliary functions that simplifythe representation of ME. For our investigations we use a potential approach, see [BP89,Jac98, StM05, HW05].

From Gauss’ law for magnetism (3.2) we deduce from Helmholtz decomposition that

there is a vector field ~A : Ωˆ I Ñ R3 such that

~B “ ∇ˆ ~A

and using Maxwell-Faraday’s law (3.3) we obtain

∇ˆˆ

~E` BBt~A

˙

“ 0.

53

Thus using Helmholtz decomposition we can conclude that there is a scalar functionϕ : Ωˆ I Ñ R such that

~E “ ´∇ϕ´ BBt~A.

This potential approach is the so-called p~A, ϕq-formulation with the vector potential ~Aand scalar potential ϕ, see [StM05]. Note there are different potential approaches and

for an overview we refer to Tabelle 2.4 in [Koc09]. The p~A, ϕq-formulation has theadvantage that the scalar potential ϕ provides a natural link to the concept of potentialdifferences which plays a crucial role in conventional simulations of electric circuits.A second advantage is that Gauss’ law for magnetism and Maxwell-Faraday’s law areautomatically fulfilled. A visual representation of all quantities is given in Figure 3.1(a).

∇∇

~B [Wbm2]

0 [Vm2]

0 [Wbm3]

~A [Wbm]

~E [Vm]

ϕ [V]ρ [Cm3]

~D [Cm2]

~Jc [Am2]

~H [Am]

0 [Am3]

BBt

BBt BBt σ [Sm]

ε [Fm]

ν [mH]

Primary Dual

∇∇

∇ BBt

(a) continuous

rCrS

q [C]

""d [C]

""j c [A]

"h [A]

0 [A]

Mε [F]

Primary Dual

ddt

ddt

ddt

G

Mν [1H]

""b [Wb]

C

Mσ [S]

S

"a [Wb]

0 [Wb]

"e [V]

φ [V]

0 [V]

ddt

(b) discrete

Figure 3.1: Tonti’s diagram or Maxwell’s house, [Ton95, Cle05, StM05].

The potential approach has a drawback: The scalar and vector potentials exhibit aso-called gauge freedom, that is, there are arbitrary in the sense that ~B and ~E are leftunchanged if the gauge transformation

~A1 “ ~A`∇χ and ϕ1 “ ϕ´ BBtχ

is applied, where the gauge function χ : Ω ˆ I Ñ R is an arbitrary scalar function, see[Jac98, HW05]. For ~B and ~E we have

~B1 “ ∇ˆ ~A1

54


“ ∇ˆ ~A`∇ˆ∇χ“ ∇ˆ ~A

“ ~B

and

~E1 “ ´∇ϕ1 ´ BBt~A1

“ ´∇ϕ´∇ BBtχ´

BBt~A` B

Bt∇χ

“ ´∇ϕ´ BBt~A

“ ~E.

A physical law which does not change under a gauge transformation is said to be gaugeinvariant. In that sense ~E and ~B are gauge invariant. To obtain a unique solution thenext step is to remove the gauge freedom of ~A and ϕ. For that reason we fix a gaugefunction except for a constant scalar field by choosing a gauge condition. In the followingwe introduce the two most common gauge conditions, namely the Coulomb gauge andLorenz gauge given by

∇ ¨ ~A “ 0 (3.9)

and

εµBBtϕ`∇ ¨

~A “ 0 (3.10)

for the case of linear, homogeneous and isotropic materials.

Remark. Lorenz gauge is named after Ludvig Lorenz. It is an invariant condition, andis often wrongly called Lorentz gauge because of confusing with Hendrik Lorentz, afterwhom Lorentz covariance is named.

To show the impact of the two gauge conditions we assume ε and µ to be constantand the functions ρ and ~Jc to be given, but related by the continuity equation (3.5), see

[MRT05]. That is, we regard ~Jc as a given source current density. Then Gauss’ law (3.1),Maxwell-Ampere’s law (3.4) and the constitutive laws (3.6) and (3.7) lead to

∆ϕ` BBt∇ ¨

~A “ ´1

ερ

∇2~A´ εµ B2

Bt2~A´∇

ˆ

∇ ¨ ~A` εµ BBtϕ˙

“ ´µ~Jc(3.11)

55

where the vector Laplace operator is denoted by ∇2. Applying Coulomb gauge to (3.11)we get the semi-decoupled system

∆ϕ “ ´1

ερ

∇2~A´ εµ B2

Bt2~A “ ´µ~Jc ` εµ BBt∇ϕ

(3.12)

consisting of an elliptic equation for ϕ and three wave equations for ~A. Applying Lorenzgauge to (3.11) we can deduce the fully decoupled system

∆ϕ´ εµ B2

Bt2ϕ “ ´1

ερ

∇2~A´ εµ B2

Bt2~A “ ´µ~Jc

(3.13)

consisting of four wave equations.

In both cases the systems lead to an unique solution if we choose initial and boundaryconditions properly, see [Eva10], and hence the gauge function χ is fixed. To derivethe system (3.12) and (3.13) we apply Coulomb and Lorenz gauge directly to (3.11).To make sure that the gauge conditions are fulfilled implicitly we need to choose theinitial and boundary conditions such that we obtain only the trivial solution for thehomogeneous wave equation

∆ψ ´ εµ B2

Bt2ψ “ 0 (3.14)

with ψ : Ω ˆ I Ñ R given by ψ “ ∇ ¨ ~A and ψ “ εµ BBtϕ ` ∇ ¨ ~A, depending on the

chosen gauge condition. We obtain (3.14) by taking the continuity equation (3.5) intoaccount. That is important since the systems (3.12) and (3.13) solve ME if and only ifthe applied gauge is implicitly fulfilled. Note that both gauges regularize the curl-curloperator in the sense that a Green function exists to determine the vector potential ~Auniquely.

For our later analysis we need to generalize the Coulomb and Lorenz gauge to obtainsuitable gauge conditons for the spatial discretization method presented in the nextsection. We rewrite Maxwell-Ampere’s law (3.4) to

~Jc “´

∇ˆ ν∇ˆ ~A´ ζ∇ξ∇ ¨ ζ~A¯

`ˆ

ε∇ BBtϕ` ζ∇ξ∇ ¨ ζ

~A

˙

` B2

Bt2ε~A

with artificial material properties ζ, ξ : Ω ˆ R Ñ R such that the SI units of ν and ζ2ξmatch. A possible class of gauge conditions reads

ϑε∇ BBtϕ` ζ∇ξ∇ ¨ ζ

~A “ 0 (3.15)

56


with ϑ P R. For ϑ “ 0 we obtain a grad-div type of Coulomb gauge

ζ∇ξ∇ ¨ ζ~A “ 0.

Moreover for ϑ “ 1 we get a type of Lorenz gauge

ε∇ BBtϕ` ζ∇ξ∇ ¨ ζ

~A “ 0.

For further discussion on gauge conditions we refer to [BCS12, StM05, Bos01, CMSW11].

Finally, ME formulated in terms of potentials using Maxwell-Ampere’s law (3.4), theconstitutive laws (3.6), (3.7) and (3.8), and the gauge condition (3.15) reads

ϑε∇ BBtϕ` ζ∇ξ∇ ¨ ζ

~A “ 0

∇ˆ ν∇ˆ ~A` ε BBt´

∇ϕ` ~Π¯

` σ´

∇ϕ` ~Π¯

“ 0

BBt~A´ ~Π “ 0

(3.16)

utilizing an auxiliary vector field ~Π : ΩˆI Ñ R3 to avoid the second-order differentiationin time for ~A.

3.1.2 Boundary and Interface Conditions

In general, EM field problems are not restricted, open boundary problems. However, forour later investigations we have to restrict ourselves to a finite domain Ω Ĺ R3. Thatis, we deal with an artificially bounded problem.

Assumption 3.2. The finite domain Ω Ĺ R3 is simply connected with the boundaryΓ “ BΩ.

In case of MQS the truncation of the domain is reasonable if a sufficiently large regionof air is around the MQS device, since the magnetic induction decays rapidly in the airtowards the boundary. As a general rule it recommends the distance from the device tothe boundary to be at least five times the radius of the device, see [CK97].

Remark 3.3. A MQS device is an EM device under the MQS assumption.

Unfortunately, this argumentation is not valid in our case since we will assume thatthe device is connected to the boundary. We assume that the main part of the deviceis sufficiently far away from the boundary and that wires with a good conductivityconnect the main part of the device to the boundary. Due to the damped wave equationscharacter of ME the fields decays towards the boundary.

To complete the system (3.16) we need boundary conditions. In addition, we haveto handle discontinuities of ε, ν and σ which can appear at the boundary between

57

different materials in our bounded domain Ω. We denote by Γint the internal boundarybetween different materials. The conditions on the internal boundary are called interfaceconditions.

We consider an internal boundary separating two different materials 1 and 2 with mate-rial properties pε1, σ1, ν1q and pε2, σ2, ν2q. Interface conditions are obtained by applyingthe Gauss’ theorem and Stokes’ theorem to ME in a small region at the internal boundarybetween two materials:

´

~E2 ´ ~E1

¯

¨ ~nq “ 0´

~B2 ´ ~B1

¯

¨ ~nK “ 0´

~D2 ´ ~D1

¯

¨ ~nK “ %´

~H2 ´ ~H1

¯

¨ ~nq “ κ

A detailed derivation is given in [Jac98, Str07]. The subscripts 1 and 2 of the EMquantities denote the quantities in materials 1 and 2. Here % describes the surfacecharge density and κ the surface current density with

%, κ : Γint Ñ R.

That is, the tangential component of ~E and the normal component of ~B are continuousfunctions across the internal boundary. The method to derive the interface conditionsis known as pill-box method.

The interface conditions motivates boundary conditions for ~E and ~B. One approach isthe so-called electric boundary condition (PEC - Perfectly Electric Conducting) and arealso called “flux wall” or “current gate” boundary conditions, see [Cle98, Ben06]. Weassume:

~E ¨ ~nq “ 0 (3.17)

~B ¨ ~nK “ 0 (3.18)

The idea is to think of a complete universe, where ME are also true outside the simulationdomain Ω. The picture is to attached a perfect conductor from outside at the boundary,where the magnetic induction does not pass through.

The next step is to interpret and motivate the boundary conditions for ~E and ~B in termsof the potentials ~A and ϕ.

Assumption 3.4. We assume that the boundary consists of k P N disjoint parts withΓ “ Ťk

i“1 Γi and for every Γi the material properties pε, σ, νq to be constant.

58


Let Assumption 3.4 be fulfilled. We examine an arbitrarily boundary part Γi. Due to(3.18) we get

0 “ ~B ¨ ~nK “ ∇ˆ ~A ¨ ~nKand we deduce that

~A ¨ ~nq “ 0 (3.19)

is a possible choice. Next we inspect (3.17) and we obtain

0 “ ~E ¨ ~nq “ ´∇ϕ ¨ ~nq ´ BBt~A ¨ ~nq “ ´∇ϕ ¨ ~nq.

To fulfill that condition a possible choice is

∇ϕ ¨ ~nq “ 0. (3.20)

The conditions (3.19) and (3.20) can be interpreted as Dirichlet boundary conditions forthe potentials.

Applying the pill-box method using the gauge conditions (3.9) and (3.10) it is possible to

show that ∇ϕ inherits the discontinuity of ~E at internal boundaries and ~A is continuous,see [AH01]. This motivates choosing homogeneous Dirichlet boundary conditions for ~Aand spatial-constant time-dependent Dirichlet boundary conditions for ϕ on each Γi. Inaddition, we choose Dirichlet boundary conditions for ~Π in accordance with ~A. Thisset of boundary conditions are a suitable link to circuit simulation, where the boundarycondition for the scalar potential ϕ can be identified with the applied potentials at thedevice contacts.

Essential for our later analysis of ME is the charge conservation expressed by the conti-nuity equation (3.5), since including EM devices into circuit models are only possible if

charges are conserved. Due to the definition of the total current ~Jt “ ~Jc ` ~Jd we obtain

ż

Γ

~Jt ¨ ~nKdF “ż

Ω

∇ ¨ ~JtdV “ 0 with ~Jt “ ∇ˆ ν∇ˆ ~A,

that is, the sum of in- and outgoing currents equals. Without loss of generality wesuppose that we number the disjoint boundary parts Γi such that the first nE ă kboundary parts have the material property σ ‰ 0 and the last k ´ nE boundary partshave the material property σ “ 0. We call Γg “ Ťk

i“nE`1 Γi the mass contact while theother Γi are called conductive contacts and we get

jg “ ´nEÿ

i“1

ji with ji “ż

Γi

~Jt ¨ ~nKdF.

59

This means that the current jg flowing through the mass contact of the EM device is thenegative sum of the currents ji, i “ 1, . . . , nE, flowing through the conductive contacts.The picture is that each conductive contact is connected to a wire from outside whilethe mass contact is grounded.

3.2 Finite Integration Technique

This section provides a survey on the finite integration technique (FIT - Finite IntegrationTechnique) for spatial discretization for solving ME in integral form. That approachwas developed and formulated by Thomas Weiland [Wei77] and is based on a stag-gered discretization. For orthogonal grids in time domain the FIT is equivalent to thefinite-difference time-domain-scheme of Kane Yee, also known as leap-frog scheme, see[Yee66].

The first step in the FIT discretization is the decomposition of the domain Ω into a finitenumber of three-dimensional volumes so that the intersection of two different volumes iseither empty - or a two-dimensional facet, a one-dimensional edge or a zero-dimensionalnode shared by both volumes. This decomposition yields a finite volumes complexG. To each edge of the volumes we prescribe an initial orientation, so that G can becharacterized as a directed graph, see Chapter B for the notation in Graph theory. Thevolume facets are supplied with an initial orientation, too.

For a rectilinear grid in Cartesian coordinates on a brick-shaped domain Ω, see [Wei77,TW96], the corresponding volumes complex G reads

G “

V pnq “ V pn pix, iy, izqq |V pn pix, iy, izqq “ rxix , xix`1s ˆ“

yiy , yiy`1

‰ˆ rziz , ziz`1s ,ix “ 1, . . . , Nx ´ 1, iy “ 1, . . . , Ny ´ 1, iz “ 1, . . . , Nz ´ 1

(

where Nx, Ny and Nz are the total numbers of (grid) nodes in x-, y- and z- direction,respectively. The total number of nodes is then N “ NxNyNz. The space indices ix, iyand iz can be reduced to one canonical space index

n “ n pix, iy, izq “ 1` pix ´ 1qKx ` piy ´ 1qKy ` piz ´ 1qKz ď N

where Kx “ 1, Ky “ Nx, Kz “ NxNy and ix “ 1, . . . , Nx, iy “ 1, . . . , Ny, iz “ 1, . . . , Nz.

To each node N pnq we associate three (grid) edges Ex pnq, Ey pnq, Ez pnq, three (grid)facets Fx pnq, Fy pnq, Fz pnq and finally, one (grid) volume V pnq.The orientation of edges and facets is given as follows: The front node of the edge Ew pnqin w-direction is N pnq. A facet Fw pnq is defined by the lower left node N pnq and thedirection w, in which its normal vector points.

Remark 3.5. The numbering scheme of the grid G introduces phantom edges, facetsand volumes at the boundary of the finite domain Ω. To not disrupt the convenientnumbering scheme, we tackle this issue later, see Subsection 3.2.4.

60


The FIT makes use of two staggered grids. The primary grid G is supported by a dualgrid rG, which is constructed by connecting the center points of neighboring primaryvolumes sharing a facet, see Figure 3.2. The center points define the dual (grid) nodesrN pnq. The definition of the dual (grid) edges rEw pnq, facets rFw pnq and volume rV pnq,are analogous to the primary grid. The orientation of dual edges and dual facets is given

primary node

dual node

dual volume

primary volume

Figure 3.2: Spatial allocation of a primary cell and a dual cell of the grid doublet, see [CW01b].

as follows: The dual back node of the dual edge rEw pnq in w-direction is rN pnq. A dual

facet rFw pnq is defined by the upper right dual node rN pnq and the direction w, in whichits normal vector points.

Remark 3.6. With this definition of the dual grid it is ensured that there is a one-to-onerelation between nodes and edges of G and volumes and facets of rG and vice versa.

The collection of all primary nodes and primary edges are denoted by N and E .

3.2.1 Maxwell’s Grid Equations

The formulation of discrete approach to electromagnetism arises from the mapping ofME in their integral form and the constitutive laws on tG, rGu. As variables of the FITwe introduce electric and magnetic voltages located on the edges defined by

"ew pnq “ż

Ewpnq

~E ¨ ~nqdE,"

hw pnq “ż

rEwpnq

~H ¨ ~nqdE,

as well as magnetic and electric fluxes and electric currents allocated at the facets definedby

""

bw pnq “ż

Fwpnq

~B ¨ ~nKdF,""

dw pnq “ż

rFwpnq

~D ¨ ~nKdF,""

j c,w pnq “ż

rFwpnq

~Jc ¨ ~nKdF.

61

The variables are tagged by arcs according to the underlying geometric object, see[Bos88]. For a convenient notation we introduce the state variable vector

"e “ p"ex p1q , . . . ,"ex pNq ,"ey p1q , . . . ,"ey pNq ,"ey p1q , . . . ,"ey pNqq

and the vector"

h,""

b ,""

d and""

j c are defined analogously. This notation allows to writeME in terms of the FIT discretization. Gauss’ law for magnetism (3.2), for example,

y

z

x

""bx pnq ""

bx pn ` Kxq

""by pnq

""by pn ` Kyq

""bz pn ` Kzq

""bz pnq

(a) Gauss’ law for magnetism.

""bz pnq "ey pn ` Kxq"ey pnq

"ex pnq

"ex pn ` Kyq

y

x

z

(b) Maxwell-Faraday’s law.

Figure 3.3: Allocation of the FIT degrees of freedom on the primary grid.

integrated over a volume V pnq, see Figure 3.3(a), can be written as

´""

bx pnq `""

bx pn`Kxq ´""

by pnq `""

by pn`Kyq ´""

bz pnq `""

bz pn`Kzq “ 0

using Gauss’ theorem. The relations for all volumes are collected in the equation

»

—

–

......

......

......

. . . ´1 1 . . . ´1 . . . 1 . . . ´1 . . . 1 . . ....

......

......

...

fi

ffi

fl

looooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooon

“S

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

...""

bx pnq""

bx pn`Kxq...

""

by pnq...

""

by pn`Kyq...

""

bz pnq...

""

bz pn`Kzq...

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

“ 0.

62


The Maxwell-Faraday’s law (3.3) integrated over a volume Fz pnq, see Figure 3.3(b),leads to

"ex pnq ´ "ex pn`Kyq ´ "ey pnq ` "ey pn`Kxq “ d

dt

""

bz pnqusing Stokes’ theorem and can be organized for all facets by

»

—

–

......

......

. . . ´1 . . . 1 . . . 1 ´1 . . ....

......

...

fi

ffi

fl

looooooooooooooooooooooooomooooooooooooooooooooooooon

“C

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

..."ex pnq

..."ex pn`Kyq

..."ey pnq

"ey pn`Kxq...

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

“ d

dt

¨

˚

˝

...""

bz pnq...

˛

‹

‚

.

The discretization of both laws exploits the numbering scheme and we refer to [Wei77,Wei96] for more details. The discretization of Gauss’ law (3.1) and Maxwell-Ampere’slaw (3.4) is analogously to the procedure described above with the only difference thatthe discrete quantities are allocated at the dual grid elements. Finally, the FIT has

primary edges

primary nodes

primary volumes

Mν

dual nodes

dual facets

primary facets

Mε, Mσ

dual edges

dual volumes

S

C

G

rGrCrS

Figure 3.4: Operator mapping.

translated ME exactly into Maxwell’s grid equations (MGE - Maxwell’s Grid Equations),[CW01b], given by

rS""

d “ q (3.21)

S""

b “ 0 (3.22)

C"e “ ´ d

dt

""

b (3.23)

rC"

h “ d

dt

""

d ` ""

j c (3.24)

with the discrete Gauss’ law (3.21), discrete Gauss’ law for magnetism (3.22), the dis-crete Maxwell-Faraday’s law (3.23) and the discrete Maxwell-Ampere’s law (3.24). The

63

discrete curl operators C and rC, the discrete divergence operators S and rS on the pri-mal and dual grid, respectively. The discrete curl operators contain only informationon the incidence relation of the volume edges and on their orientation, see Figure 3.5.The divergence operators collect information on the incidence relation and on the ori-entation of the facets of the volumes. The curl operator maps from edges to facets andthe divergence operator from facets to volumes, see Figure 3.4. The unknowns are thediscrete electric field and magnetic field "e,

"

h : I Ñ R3N , discrete electric and magneticinduction

""

d,""

b : I Ñ R3N , conduction current""

j c : I Ñ R3N and distribution of chargesq : I Ñ RN . Tables of SI units are given in Table 3.4 and 3.5.

1

´1

1

´1

Figure 3.5: Orientation of the curl.

Remark 3.7. The discrete distribution of charges q is located on dual volumes andhence q should be written as

"""q to be consistent with the notation. Nonetheless, forclarity, we simple write q in abuse of notation.

So far the discretization of the physical laws does not require any approximation sincethe ME have been directly applied to the grid by using topological information only.For a complete discretization of ME the constitutive laws (3.6), (3.7) and (3.8) have tobe related to the discrete EM quantities allocated at the grid doublet. At this point all

quantity geometric object SI units"e primary edges V""

d dual surfaces C“As""

b primary surfaces Wb“Vs"

h dual edges A""

j c,""

j t dual surfaces Aφ primary node V"a primary edges Wb“Vs"π primary edges Vq dual volumes C“As

Table 3.4: Discrete field quantities.

operator SI unitsddt

1sG, rG 1

C, rC 1

S, rS 1

Table 3.5: Discrete differential operators.

64


metric information enters the spatial discretization and the constitutive laws establisha coupling between the primary and the dual EM quantities. Now the need for the griddoublet becomes clear. For example the discrete version of the constitutive laws (3.6)

needs to relate "e and""

d, but these discrete quantities are defined on different geometricobjects. We can relate them because of the one-to-one relation between edges of G andfacets of rG.

σ pn Kx Kyqε pn Kx Kyq

σ pnqε pnqσ pn Kxqε pn Kxq

σ pn Kyqε pn KyqN pn Kx Kyq N pn Kyq

N pnqN pn KxqEz pnq

rFz pnqy

x

z

(a) permittivity and conductivity

Fy pnqrEy pnq

N pnq

N pn Kyqν pn Kyq

ν pnq

y

x

z

(b) reluctivity

Figure 3.6: Material properties located on the grid.

Assumption 3.8. The material properties are constant in each primary volume.

Let Assumption 3.1 and 3.8 be true. To derive a discrete version of the constitutive laws(3.6) for linear, inhomogeneous and isotropic permittivities we employ the rectangle rule.

Regarding the edge Ez pnq and the facet rFz pnq. Using the midpoint rectangle rule weget

"ez pnq “ |Ez pnq| |~E|z,n `O`

h3˘

(3.25)

where |~E|z,n is the sample value of the electric field at the midpoint of the edge Ez pnq,|Ez pnq| is the edge lengths and

h “ maxwPtx,y,zu1ďnďN

|Ew pnq|

is the maximum length of the edges. The discontinuities of electric induction ~D atinternal boundaries in normal direction does not effect the discretization since we need toswitch to the electric field ~E for the discretization of the constitutive law (3.6). Applying

65

quantity SI units

Mε F=AsVMν 1H“AVsMσ S“AV

Table 3.6: Discrete material matrices.

the top-left and top-right rectangle rule we get""

dz pnq “ rεz pnq |~E|z,n `O`

h3˘

(3.26)

with the average permittivity

rεz pnq “ 1

4pε pnxyq |Fz pnxyq| ` ε pnxq |Fz pnxq| ` ε pnq |Fz pnq| ` ε pnyq |Fz pnyq|q

and nxy “ n´Kx ´Ky, nx “ n´Kx, ny “ n´Ky. Note that for the dual facet rFz pnqit holds

ˇ

ˇ

ˇ

rFz pnqˇ

ˇ

ˇ“ 1

4p|Fz pnxyq| ` |Fz pnxq| ` |Fz pnq| ` |Fz pnyq|q

and ε pnq denotes the permittivity on the volume V pnq, see Figure 3.6(a). Combining(3.25) and (3.26) yields

""

dz pnq “ εz pnq"ez pnq `O`

h3˘

.

with εz pnq “ rεzpnq|Ezpnq|

. Finally we get the permittivity matrix

Mε “ diag pεx p1q , . . . , εx pNq , εy p1q , . . . , εy pNq , εz p1q , . . . , εz pNqq .The conductivity matrix Mσ for linear, inhomogeneous and isotropic conductivities is de-fined analogously, see [Cle98, Kru00, Ben06]. On a similar way a linear, inhomogeneousand isotropic reluctivity matrix can be deduced by taking Figure 3.6(b) into account. For

the derivation of a nonlinear, inhomogeneous and anisotropic reluctivity matrix Mνp""

bqgiven by Brauer’s model, we refer to [Sch11]. The discrete constitutive laws reads:

""

d “ Mε"e (3.27)

""

j c “ Mσ"e (3.28)

"

h “ Mνp""

bq""

b (3.29)

Table 3.6 shows the SI units.

Remark 3.9. The discrete material matrix of permittivities is diagonally positive defi-nite, while the discrete material matrix of conductivities is typically diagonally positivesemi-definite if insulators are present in our domain Ω, otherwise positive definite. Incase of non-orthogonal grids band structured matrices results. For Brauer’s model thediscrete material matrix of reluctivities is positive definite.

66


The basic approximation of the constitutive laws leads to a staircase approximations atcurved boundaries. In practice this limitation is overcome by subgridding at boundariesor using other more elaborate schemes, see [TW96, Cle05].

3.2.2 Algebraic Properties of the Discrete Operators

The discrete operators in terms of the FIT have several important inheritances of theircontinuous counterparts and are composed of simple two-banded matrices, which can beinterpreted as discretized partial differential operators, see [BDD`92, CW01b].

Let be w P tx, y, zu. We introduce the upper shift matrices Uw P t0, 1uNˆN with

pUwqij “ δi`Kw,j and Uw “ UKwx . (3.30)

We define the discretized partial differential operators Pw P t´1, 0, 1uNˆN by

Pw “ Uw ´ I

where Pw is nonsingular, w P tx, y, zu. The discrete curl operators can be written as

C “»

–

0 ´Pz Py

Pz 0 ´Px

´Py Px 0

fi

fl P t´1, 0, 1u3Nˆ3N

and the duality of the two grids yields the simple relation

rC “ CJ.

The discrete divergence operators are constructed by

S “ “

Px Py Pz

‰ P t´1, 0, 1uNˆ3N and rS “ “´PJx ´PJy ´PJz‰

.

Finally the discrete gradient operators are obtained by

G “ ´rSJ and rG “ ´SJ,

see [CW01b, CW01a].

Lemma 3.10 (Lemma A.1., [Sch11]). Let be v, w P tx, y, zu. The relation

PvPw “ PwPv

holds true.

Proof . Straightforward calculus using (3.30) leads to

PvPw “ pUv ´ Iq pUw ´ Iq“ UvUw ´ Uv ´ Uw ` I

67

“ UKvx UKw

x ´ Uv ´ Uw ` I

“ UKwx UKv

x ´ Uv ´ Uw ` I

“ UwUv ´ Uw ´ Uv ` I

“ pUw ´ Iq pUv ´ Iq“ PwPv.

The result reflects the interchange of partial derivatives as in the continuous case,[BDD`92].

Lemma 3.11 ([BDD`92]). The discrete operator identities

SC “ 0 rSrC “ 0 (3.31)

CG “ 0 rCrG “ 0

hold true.

Proof . To prove the identities we use Lemma 3.10. We get

SC “ “

PyPz ´ PzPy PzPx ´ PxPz PxPy ´ PyPx

‰ “ 0.

The dual case is analogous. To show the other identities we simply transpose (3.31).

That is, the discrete gradient, curl and divergence inherit important operator identitiesfrom their continuous counterparts, namely

∇ ¨∇ˆ ” 0 and ∇ˆ∇ ” 0,

which is an important property of the FIT discretization.

We have already seen that the continuity equation can be derived from ME. Due to theproperties of the discrete operators given in Lemma 3.11 that is possible in the discretecase, too. From (3.24) we derive the built-in discrete continuity equation by

d

dtrS

""

d ` rS""

j c “ 0 (3.32)

which corresponds to the continuous counterpart and is an essential feature of FIT.The discrete continuity equation is of great importance for our later investigations incircuit models including EM devices. The discrete continuity equation ensures that noerroneous charges arises, see [CW01b].

Lemma 3.12. For the discrete operators the relations

ker S “ im C kerrS “ im rC

ker C “ im G ker rC “ im rG

hold true.

68


Proof . We apply Lemma 3.10 and 3.11. Exemplarily we show ker S “ im C. Due toLemma 3.11 we achieve directly ker S Ą im C. Let be w P ker S. Then we get

Pxw1 ` Pyw2 ` Pzw3 “ 0 ô w1 “ ´P´1x pPyw2 ` Pzw3q .

Next we choose u1 “ 0, u2 “ P´1x w3 and u3 “ ´P´1

x w2. So we obtain w “ Cu and hencew P im C. The other relations can be deduced in a similar way.

3.2.3 Discrete Potential Formulation and Gauge Conditions

In analogy to ME we introduce auxiliary functions to simplify the representation of MGEand use a discrete potential approach, see [Cle98, CW99, MMS01].

From discrete Gauss’ law for magnetism (3.22) we deduce from a discrete version ofHelmholtz decomposition that there is a vector function "a : I Ñ R3N such that

""

b “ C"a (3.33)

and using discrete Maxwell-Faraday’s law (3.23) we obtain

C

ˆ

"e ` d

dt"a

˙

“ 0

and conclude, using a discrete version of Helmholtz decomposition, that there is a vectorfunction φ : I Ñ RN such that

"e “ ´Gφ´ d

dt"a, (3.34)

see [Cle05]. This approach is the discrete p"a, φq-formulation with the discrete vectorpotential "a and discrete scalar potential φ. It fulfills immediately the discrete Gauss’law for magnetism and the discrete Maxwell-Faraday’s law because important propertiesfrom vector calculus are transfered to the discrete level, see Lemma 3.11. A visualrepresentation of all quantities is given in Figure 3.1(b).

As in the continuous case we need a gauge condition to remove the gauge freedom sincethe discrete curl-operator inherits the non-uniqueness from its continuous counterpart. Acommon gauge condition approach is the grad-div regularization, [CW02], which utilizesthe discrete gradient and divergence operator and suitable discrete artificial materialmatrices. This motivates a new discrete class of gauge conditions in terms of the FITgiven by

ϑMεGd

dtφ`MζGMξ

rSMζ"a “ 0 (3.35)

where the artificial material matrices Mζ maps primary edges to dual facets, Mξ mapsdual points to primary volumes and ϑ P R is a “slider” between a type of discrete

69

Coulomb and Lorenz gauge. The discrete class of gauge conditions (3.35) is the discreteanalogon to (3.15). The discrete material matrix Mξ is called norm matrix and suppliesthe correct units to the discrete grad-div regularization. In case of ϑ “ 0 suitable choicesfor the discrete material matrices Mζ and Mξ are discussed in [CW02, Cle05, BCDS11].For another type of discrete gauge conditions motivated by damped wave equations werefer to [BCS12].

Assumption 3.13. The discrete artificial material matrices Mζ and Mξ are positivedefinite.

Let Assumption 3.13 be true. For ϑ “ 0 we obtain a type of discrete Coulomb gauge

rSMζ"a “ 0

due to rSMξG is nonsingular. The discrete Coulomb gauge is known from literature.Moreover, for ϑ “ 1 we obtain a type of discrete Lorenz gauge

MεGd

dtφ`MζGMξ

rSMζ"a “ 0

and selecting in addition Mζ “ Mε yields

d

dtφ`Mξ

rSMε"a “ 0,

due to rSMεG is nonsingular. For linear, homogeneous and isotropic materials and anequidistant grid the discrete grad-div regularization regularizes the discrete curl-curlmatrix and the resulting discrete operator corresponds to the discrete vector Laplacian.

Lemma 3.14. Let M P RNˆN be positive definite. Then, the matrix

C “ rCMνC´GJMrS

is positive definite.

Proof . We use the relation G “ ´rSJ. The matrix C is symmetric positive semidefinitesince C is the sum of two positive semidefinite matrices. To show positive definitenesswe prove the nonsingularity. Let be x P ker C. Then

´

rCMνC` rSJMrS¯

x “ 0 ô Cx “ 0 and rSx “ 0,

see Lemma A.3. Hence x P ker CX kerrS. With Lemma 3.12 it is clear that

x P ker CX im rC “ ker CX pker CqK

and hence x “ 0.

Note that rCMνC`MζrSJMξ

rSMζ is not necessarily positive definite. Hence not an arbi-trary type of discrete Coulomb or Lorenz gauge leads to a gauge condition in the senseof a discrete curl-curl operator regularization.

70


3.2.4 Phantom Objects and Discrete Boundary Conditions

The numbering scheme of the grid G introduces needless phantom objects at the bound-ary. The phantom objects are edges, facets and volumes which have to be disregarded.In order to disregard those objects we follow and extend the idea of [Doh92, Sch11].Table 3.7 gives an overview of the number of non-phantom objects.

object number of non-phantom objects

primary nodes/dual volumes NxNyNz

primary facets/dual volumes pNx ´ 1qNyNz

`Nx pNy ´ 1qNz

`NxNy pNz ´ 1qprimary facets/dual edges Nx pNy ´ 1q pNz ´ 1q

` pNx ´ 1qNy pNz ´ 1q` pNx ´ 1q pNy ´ 1qNz

primary volumes/dual nodes pNx ´ 1q pNy ´ 1q pNz ´ 1q

Table 3.7: Number of non-phantom objects.

Example 3.15. Regarding the primary FIT grid of two points in each direction asshown in Figure 3.7. The grid consists of 8 nodes, 12 edges, 6 facets and one volume.The numbering scheme introduces 12 edges, 18 facets and 7 volumes which are needless.

y

x

z

Ey p5q Ey p6q

Ey p1q Ey p2q

Ez p4q

Ez p1q Ez p2qEx p5q

Ex p7q

Ex p1q

Ex p3qEz p3q

(a) non-phantom edges

y

x

z

Fy p1q

Fy p3q

Fx p1q Fx p1qV p1qFz p1q

Fz p5q

(b) non-phantom facets and volumes

Figure 3.7: Primary FIT grid of dimensions 2ˆ 2ˆ 2 with non-phantom objects.

To find all phantom objects it is sufficient to characterize the phantom edges. Theseedges are always attached to points on the boundary that are addressed by n pix, iy, izq,

71

where one direction reaches its maximum iw “ Nw with w P tx, y, zu. For each w Ptx, y, zu the set

Hw “ t1 ď n pix, iy, izq ď N |iw “ Nwucontains the indices of all points with an attached phantom edge in direction w and wefade-out all phantom objects using the diagonal fade-out matrix Fw P t0, 1uNˆN givenby:

pFwqij “#

1, for i “ j and i R Hw,

0, else.

The sets Hxyz “ Hx YHy YHz and Hvw “ Hv YHw contain all points connected to atleast one phantom edge and connected to at least one phantom edge in the pv-wq-planewith v, w P tx, y, zu, v ‰ w, respectively. Next we investigate some properties of thefade-out matrices.

Lemma 3.16 (Lemma A.3., [Sch11]). The matrices Fw are orthogonal projectors andfor v ‰ w

FwFv “ FvFw, (3.36)

FwFvPv “ FvPvFw (3.37)

is valid with v, w P tx, y, zu.Proof . The projector properties of Fw as well as (3.36) are clear since they are diagonalmatrices containing only zeros and ones. The left-hand side of (3.37) reads:

pFwFvPvqij “

$

’

&

’

%

´1, for j “ i and i R Hwv,

1, for j “ i`Kv and i R Hwv,

0, else.

The right-hand side of (3.37) reads:

pFvPvFwqij “

$

’

&

’

%

´1, for j “ i and i R Hwv,

1, for j “ i`Kv, j R Hw and i R Hv,

0, else.

Now we show that both sides equals. Since i R Hv we can write

i “ ix ` iyKy ` izKz, with iv ă Nv

and thus j “ i`Kv gives

j “ jx ` jyKy ` jzKz, with jv “ iv ` 1 ď Nv.

Then we know that jw “ iw since v ‰ w and thus the condition i R Hw is equivalent toj “ i`Kv R Hw for v ‰ w.

72


All points addressed by the numbering scheme are included in the primary grid but onlysubsets of the addressed edges, facets and volumes really exist. Only edges not in Hw,w P tx, y, zu, exists. Furthermore facets and volumes exists if and only if all their edgesexist. To fade-out the phantom objects we define

FN “ I, FE “»

–

Fx 0 00 Fy 00 0 Fz

fi

flFF, “»

–

FyFz 0 00 FxFz 00 0 FxFy

fi

fl and FV “ FxFyFz

where FN, FE, FF and FV denote the fade-out projectors for all points, edges, facets andvolumes in the primary grid. Analogously we define the corresponding counterparts forthe dual grid and benefit of the relation between both grids.

Corollary 3.17. The fade-out projectors FN, FE, FF and FV are orthogonal projectors.

Next we define the discrete operators with fade-out phantom objects. The gradientoperator maps points to edges and we have to ignore contributions from phantom pointsand edges. We achieve

GF “ FEGFN and rGF “ FFrGFV.

The curl operator maps edges to facets and therefore we have to ignore contributionsfrom phantom edges and facets. We get

CF “ FFCFE and rCF “ FErCFF.

In the end the divergence operator maps facets to volumes and hence we have to ignorecontributions from phantom facets and volumes. We gain

SF “ FVSFF and rSF “ FNrSFE.

All discrete operators with fade-out phantom objects have a redundancy.

Lemma 3.18 (Corollary A.5., [Sch11]). For the discrete operator with fade-out phantomobjects the relations

GF “ FEG rGF “ rGFV

CF “ FFC rCF “ rCFF

SF “ FVS rSF “ rSFE

hold true.

Proof . This is a consequence of Lemma 3.16.

For the discrete operators with fade-out phantom objects all important properties stillhold true.

73

Lemma 3.19 (Theorem A.6., [Sch11]). The discrete operator identities

SFCF “ 0 rSFrCF “ 0

CFGF “ 0 rCFrGF “ 0

hold true.

Proof . This is a consequence of Lemma 3.11 and 3.18.

Remark 3.20. Sometimes in literature the discrete partial differential operators aredirectly constructed as Pw “ FwPw with w P tx, y, zu, for example, as in [Ben06].

Not only in the discrete operators the phantom objects occur but also in the discretematerial matrices. The discrete permittivity matrix as well as the discrete conductivitymatrix are mapped from primary edges to dual facets and we get

MFε “ FEMεFE and MF

σ “ FEMσFE.

Furthermore, the discrete reluctivity matrix maps from primary facets to dual edges andreads

MFν “ FFMµFF.

In fact we do not simply want to fade-out the phantom objects but we want to delete thecorresponding rows and columns within the discrete operators, too. For that we extendthe idea of phantom objects of [Sch11] and we define the matrices

Dw P t0, 1uN´|Hw|ˆN , Dvw P t0, 1uN´|Hvw|ˆN , Dxyz P t0, 1uN´|Hxyz |ˆN

related to the fade-out projectors by

DJwDw “ Fw DwDJw “ I

DJvwDvw “ FvFw DvwDJvw “ I

DJxyzDxyz “ FxFyFz DxyzDJxyz “ I

with v, w P tx, y, zu, v ‰ w. We construct the deletion and shrinking matrices

DN “ I, DE “»

–

Dx 0 00 Dy 00 0 Dz

fi

fl , DF “»

–

Dyz 0 00 Dxz 00 0 Dxy

fi

fl and DV “ Dxyz

where DN, DE, DF and DV denotes the matrix deleting all rows of FN, FE, FF and FV

belonging to phantom objects. For the deletion matrices the relations

DJNDN “ FN DNDJN “ I

DJEDE “ FE DEDJE “ I

74


DJFDF “ FF DFDJF “ I

DJVDV “ FV DVDJV “ I

hold true.

Now we remove the phantom objects. For that we left-multiply the different discretizedlaws of MGE by the corresponding deletion matrix and we set the unknowns correspond-ing to phantom objects to zero. From MGE (3.21) to (3.24) we deduce the phantom-freeMGE

rSD

""

dD “ qD (3.38)

SD

""

bD “ 0 (3.39)

CD"eD “ ´ d

dt

""

bD (3.40)

rCD"

hD “ d

dt

""

dD ` ""

j c,D (3.41)

where the existing unknowns are

"eD “ DE"e,

""

dD “ DE

""

d,""

j c,D “ DE

""

j c,"

hD “ DF"

h,""

bD “ DF

""

b and qD “ DNq

and for the phantom-free operators the relations

rCD “ DErCDJF , rSD “ DN

rSDJE, SD “ DVSDJF and CD “ DFCDJE

hold true. Applying the same deduction as above, the discrete constitutive laws (3.27),(3.28) and (3.29) yields

""

dD “ MDε

"eD (3.42)""

j c,D “ MDσ

"eD (3.43)"

hD “ Muνp

""

bDq""

bD (3.44)

where the phantom-free material matrices are given by

MDε “ DEMεD

JE, MD

σ “ DEMσDJE and MDν “ DFMνD

JF .

The discrete equations for the vector and scalar potential (3.33) and (3.34) result in

""

bD “ CD"aD (3.45)

"eD “ ´GDφD ´ d

dt"aD (3.46)

at which the existing unknowns are

φD “ DNφ and "aD “ DE"a

and the phantom-free gradient reads GD “ DEGDJN.

75

Remark 3.21. By EF Ă E we denote the set of non-phantom edges indices. We caninterpret GD as the transpose incidence matrix of the directed graph pN , EFq. We willcome back to that later when motivating the boundary conditions.

For completeness we have rGD “ DFrGDJV. The phantom-free operators still have the

following properties:

Lemma 3.22. The phantom-free operators identities

SDCD “ 0 rSDrCD “ 0

CDGD “ 0 rCDrGD “ 0

hold true.

Proof . This is a consequence of Lemma 3.19.

Using (3.41) and Lemma 3.22 we can derive the phantom-free continuity equation givenby

d

dtrSD

""

dD ` rSD

""

j c,D “ 0 (3.47)

but note that we can derive (3.47) also directly from (3.32). Therefore charge conserva-tion is also fulfilled for the phantom-free operators.

From the phantom-free equation for the vector potential (3.45) and scalar potential (3.46)we can deduce that the phantom-free Gauss’ law for magnetism (3.39) and Maxwell-Faraday’s law (3.40) are fulfilled automatically like in the continuous case. Based on(3.47) we conclude that if

rSD

""

dD pt0q “ qD pt0q , t0 P I,then the phantom-free Gauss’ law (3.38) is always fulfilled like in the continuous case.That is, it is sufficient to take the phantom-free Maxwell-Ampere’s law (3.41) into ac-count using the phantom-free potential formulation.

Boundary Conditions

The next step is to incorporate the boundary conditions. Here we focus on the PECcase and apply Dirichlet boundary conditions for the unknowns.

Let ΩN “ t1, . . . , Nu be the set of node indices and ΓN “ tn P ΩN|N pnq P Γu the setof boundary node indices. We denote by ΩC

N “ ΩNzΓN the set of non-boundary nodeindices and nφ “

ˇ

ˇΩCN

ˇ

ˇ. To describe the non-phantom edge indices properly we needsome notation. Let

Ex “ tn P N|n P Hxu Ey “ tn`N P N|n P Hyu Ez “ tn` 2N P N|n P Hzu

76


and

Ex “ t1, . . . , Nu zEx Ey “ tN ` 1, . . . , 2Nu zEy Ez “ t2N ` 1, . . . , 3Nu zEzbe the index sets of phantom and non-phantom edges in each direction. Then the set ofnon-phantom edges indices is given by EF “ ExY Ey Y Ez. Let ΩE “ t1, . . . , |EF|u be theset of the renumbered non-phantom edge indices. For renumbering the non-phantomedges we define the injective and surjective mapping

p : t1, . . . , 3Nu Ñ ΩE

with the property

i ă j ô p piq ă p pjq , i, j P t1, . . . , 3Nu ,where for the preimage

p´1 pkq P EF, k P ΩE

holds true. By

ΓE “

n P ΩE | m “ p´1 pnq ,m P Ew and Em pwq Ă Γ, w P tx, y, zu(

we denote the set of renumbered boundary edge indices and ΩCE “ ΩEzΓE is the set

of renumbered non-boundary non-phantom edge indices, where the non-boundary non-phantom edges are degrees of freedom. We denote na “

ˇ

ˇΩCE

ˇ

ˇ.

We introduce the unknown and boundary selection matrices

UN P t0, 1unφˆN , UE P t0, 1unaˆ|EF| , BN P t0, 1u|ΓN|ˆN , BE P t0, 1u|ΓE|ˆ|EF|

for nodes and edges defined by

UJNUN “ UF,N UNUJN “ I BJNBN “ BF,N BNBJN “ I

UJEUE “ UF,E UEUJE “ I BJEBE “ BF,E BEBJE “ I

with the properties

UF,N ` BF,N “ FN and UF,E ` BF,E “ FE.

With that we obtain the relations

UN “ UNUF,N UJN “ UF,NUJN BN “ BNBF,N BJN “ BF,NBJN UNBF,N “ 0

and

UE “ UEUF,E UJE “ UF,EUJE BE “ BEBF,E BJE “ BF,EBJE UEBF,E “ 0

hold true

At this point we need the orthogonality of each grid complex G and rG.

77

Remark 3.23. For every diagonal matrix D P R|EF|ˆ|EF| we get UEDBJE “ 0.

Now we incorporate the boundary conditions into the equations. We start with thephantom-free constitutive laws. Left-multiplying (3.42) by UEDJE and using Remark 3.23we acquire on the one hand

UEDJE""

dD “ UEDJEMDε

"eD

“ UEDJEDEMεDJEDE

"e

“ UEFEMεFE"e

“ UEMε pUF,E ` BF,Eq"e

“ UEMεUF,E"e

“ UEMεUJEUE

"e

and on the other hand

UEDJE""

dD “ UEDJEDE

""

d

“ UEFE

""

d

“ UE

""

d.

With that the phantom-free constitutive laws (3.42), (3.43) and (3.44) yields the reduceddiscrete constitutive laws

""

du “ Muε

"eu, (3.48)""

j c,u “ Muσ

"eu, (3.49)"

hu “ Muνp

""

buq""

bu (3.50)

with the unknowns

"eu “ UE"e,

""

bu “""

bD,"

hu “ "

hD,""

du “ UE

""

d and""

j c,u “ UE

""

j c

and reduced discrete material matrices

Muε “ UEMεU

JE, Mu

σ “ UEMσUJE and Muνp

""

buq “ MDν p

""

bDq.From phantom-free Maxwell-Ampere’s law (3.41) we get the reduced discrete Maxwell-Ampere’s law

rCu"

hu “ d

dt

""

du ` ""

j c,u (3.51)

with the reduced discrete dual curl operator

rCu “ UErCDJF .

78


From the equations of the phantom-free vector and scalar potential (3.45) and (3.46) weobtain

""

bu “ Cu"au ` Cb

"ab (3.52)

"eu “ ´Guφu ´Gbφb ´ d

dt"au (3.53)

at which the unknowns read

φu “ UNφ, φb “ BNφ,"au “ UE

"a and "au “ BE"a

and reduced discrete operators are given by

Gu “ UEGUJN, Gb “ UEGBJN, Cu “ DFCUJE and Cb “ DFCBJE.

In addition the reduced discrete dual divergence operator reads

rSu “ UNrSUJE

We show that important reduced discrete operator identities are still valid.

Lemma 3.24. The relations Gu “ ´rSJu and rCu “ CJu hold true.

Lemma 3.25. The reduced discrete operator identity rSurCu “ 0 and CuGu “ 0 hold

true.

Proof . We infer from Gu “ UEGUJN that UJEGuUN “ UF,EGUF,N. For every edge in ΓE

the front and back node are in ΓN. That is, UF,N set exactly that columns to zero whichare not effected by UF,E Hence GUF,N “ UF,EGUF,N and GUJN “ UF,EGUJN. We get

rSurCu “ UN

rSUJEUErCDJF

“ UNrSUF,E

rCDJF

“ UNrSrCDJF

“ 0,

see Lemma 3.11. The other statement follows directly.

Using (3.51) and Lemma 3.25 we can derive the reduced discrete continuity equationgiven by

d

dtrSu

""

du ` rSu

""

j c,u “ 0 (3.54)

but note that we can (3.54) derive also directly from (3.47). Left-multiplying (3.47) byUNDJN we obtain

0 “ d

dtUNDJNrSD

""

dD ` UNDJNrSD

""

j c,D

79

“ d

dtUNDJNDN

rSDJEDE

""

d ` UNDJNDNrSDJEDE

""

j c

“ d

dtUNFN

rSFE

""

d ` UNFNrSFE

""

j c

“ d

dtUN

rS pUF,E ` BF,Eq""

d ` UNrS pUF,E ` BF,Eq""

j c

“ d

dtUN

rSUF,E

""

d ` UNrSUF,E

""

j c

“ d

dtUN

rSUJEUE

""

d ` UNrSUJEUE

""

j c

“ d

dtrSu

""

du ` rSu

""

j c,u

using BF,EGUJN “ 0 since UF,N sets exactly that columns to zero which are effected byBF,E. Therefore charge conservation is also fulfilled.

We already mentioned that GD can be interpreted as the transpose incidence matrix ofthe directed graph pN , EFq, see Remark 3.21. In fact GJ

u is a kind of reduced incidencematrix of the directed graph pN , EFq with more then one reference node, due to settingDirichlet boundary conditions for all boundary nodes and edges. That is an importantobservation for the later index analysis of the resulting DAE from MGE.

Remark 3.26. The reduced discrete operator Gu has full column rank.

3.2.5 Maxwell’s Grid Equations with Boundary Excitation

In this subsection we formulate a new class of reduced discrete gauge conditions in termsof FIT for the non-phantom and non-boundary unknowns. We describe the boundaryconditions for the scalar potential as excitation of the EM fields at the boundary andformulate the current through the EM devices. The excitation and current formulationplay a vital role for the circuit models including EM devices.

Motivated by (3.35) we reformulate the class of discrete gauge conditions into the classof reduced discrete gauge conditions given by

ϑMuεGu

d

dtφu `Mu

ν"au “ 0 (3.55)

with reduced discrete artificial material matrices given by

Muν “ Mu

ζGuMuξrSuMu

ζ , Muζ “ UEMζU

JE and Mu

ξ “ UNMξUJN.

Note, we cannot deduce (3.55) directly from (3.35) due to the presence of the boundaryconditons but (3.55) is motivated by that. For later investigation we left-multiply (3.55)

by rSu and we regard

ϑrSuMuεGu

d

dtφu ` rSuMu

ν"au “ 0. (3.56)

80


The next step is to describe the Dirichlet boundary conditions for the discrete boundaryscalar potentials given by φb : I Ñ R|ΓN| in more detail to have an excitation for theEM fields at the domain boundary. Here we do not follow the approach presentedin [DHW04, Ben06, BBS11, Sch11], where the excitation is constructed using differentconductor models and applied as a source term. In Subsection 3.1.2 we motivate spatial-constant time-dependent Dirichlet boundary conditions for scalar potentials on each ΓN,i

with

ΓN,i “ tn P ΩN|N pnq P Γiu .Without loss of generality we suppose that the first nE ă k boundary parts ΓN,i are

conductive contacts and ΓN,g “ Ťki“nE`1 ΓN,i is the mass contact. That is, the EM

device has nE`1 contacts. At the mass contact we apply zero potential. The potentialsat the conductive contacts are described by vE : I Ñ RnE . Next we construct anpre-excitation matrix X P RNˆnE defined by

pXqij “#

1, if i P ΓN,j,

0, else.

which maps from conductive contacts to nodes, acting only on boundary nodes at con-ductive contacts. Note that we directly skip the mass contact because of the zero po-tential. We write the boundary conditions in terms of the input function and obtain theboundary excitation

φb “ BNXvE.

With the pre-excitation matrix we define excitation matrix Λu P RnaˆnE by

Λu “ ´GbBNX (3.57)

which maps from conductive contacts to non-phantom and non-boundary edges. Due tothat construction it is obvious that the excitation matrix acts only on edges attached toconductive contacts and Λu has full column rank.

Assumption 3.27. We assume homogeneous Dirichlet boundary condition for the dis-crete vector potential, that is, "ab “ 0, and that the applied potential at the mass contactis zero.

Let Assumption 3.27 be true. The applied potential at the EM device conductive con-tacts generates currents and the reduced discrete total current density in terms of FITis given by reduced discrete Maxwell-Ampere’s law (3.51). We get

""

j t,u “ d

dt

""

du ` ""

j c,u

“ rCuMuνpCu

"auqCu"au.

81

For our later investigation we are interested in the reduced discrete total current densityat the conductive contacts. Here we can utilize the excitation matrix Λu. The reduceddiscrete total current density at a contact is the sum of all contributions from non-phantom and non-boundary edges attached to the contact taking the edges orientationinto account. That is, the reduced discrete total current density at the conductivecontacts can be described by

jE “ ΛJu rCuMuνpCu

"auqCu"au P RnE . (3.58)

Assumption 3.28. For a consistent contact formulation we assume:

(i) There are at least two conductive contacts.

(ii) The conductive contacts are disjoint and simply connected.

(iii) Between two conductive contacts are at least two primary surfaces.

To ensure this we need a sufficiently fine spatial discretization of the EM device.

To show that we have a consistent discrete contact current formulation we formulate thefollowing lemma.

Lemma 3.29. Let Assumption 3.28 be true. The matrix CuΛu has full column rank.

Proof . Regarding the j-th column of CuΛu. The i-th row of the j-th column of CuΛu

is nonzero if the i-th primary facets consists of one or three primary edges connectedwith the j-th conductive contact. Such a primary facet always exists for each conductivecontact. If the i-th row of the j-th column is nonzero the other columns are zero at thei-th row.

Next we derivative the reduced curl-curl equation. Starting with the reduced discreteMaxwell-Ampere’s law (3.51), the reduced discrete constitutive laws (3.48), (3.49) and(3.50), using the excitation matrix (3.57) and formulated in terms of the reduced poten-tials (3.52) and (3.53) we gain

0 “ rCu"

hu ´ d

dt

""

du ´ ""

j c,u

“ rCuMuνp

""

buq""

bu ´Muε

d

dt"eu ´Mu

σ"eu

“ rCuMuνpCu

"auqCu"au `Mu

ε

d

dt

ˆ

Guφu `Gbφb ` d

dt"au

˙

`Muσ

ˆ

Guφu `Gbφb ` d

dt"au

˙

and we are ending with the reduced curl-curl equation

0 “ MuεGu

d

dtφu `Mu

ε

d2

dt2"au `Mu

σGuφu ` rCuMuνpCu

"auqCu"au `Mu

σ

d

dt"au

´MuεΛu

d

dtvE ´Mu

σΛuvE.

(3.59)

82


Grouping a reduced discrete gauge conditions given by (3.56), the reduced discrete totalcurrent density at the conductive contacts (3.58) and the reduced curl-curl equation(3.59) we obtain the MGE

jE ´ ΛJu rCuMuνpCu

"auqCu"au “ 0

ϑrSuMuεGu

d

dtφu ` rSuMu

ν"au “ 0

MuεGu

d

dtφu `Mu

ε

d

dt"πu `Mu


"auqCu"au `Mu

σ"πu

´MuεΛu

d

dtvE ´Mu

σΛuvE “ 0

d

dt"au ´ "πu “ 0

(3.60)

with the auxiliary vector "πu to avoid the second-order differentiation in time for "au. Thenumber of non-boundary nodes is nφ, the number of non-boundary edges is na, nπ andthe number of conductive contacts by nE. The given vector function vE ptq describes theapplied potential at the conductive contacts in time t, I “ rt0, T s Ă R. The unknownsare the (reduced) discrete scalar potentials φu : I Ñ Rnφ , the (reduced) discrete vectorpotentials "au : I Ñ Rna , the auxiliary vector "πu : I Ñ Rnπ and the current jE : I Ñ RnE

through the conductive contacts.

Remark 3.30. Let pφu,"au,

"πuq P Rnφ ˆRna ˆRnπ be a solution of (3.60). Then all fieldquantities can be derived. We obtain

""

bu “ Cu"au,

"eu “ ´Guφu ` ΛuvE ´ "πu,""

du “ Muε

"eu,""

j c,u “ Muσ

"eu,"

hu “ Muνp

""

buq""

bu

and

qu “ rSu

""

du.

3.2.6 Numerical Analysis of Maxwell’s Grid Equations

In this subsection we investigate MGE (3.60) using the Coulomb and Lorenz gaugewithout the current equation since it is only an explicit function evaluation. We aremainly interested in the index of the resulting DAEs. We obtain similar results as[BCS12] but we do not use the differentiation index, the excitation of the fields is comingfrom boundary conditions instead of source term and we regard a different class of gaugeconditions.

83

We collect some basic assumptions and properties for the discrete operators from theprevious section.

Assumption 3.31. The reduced discrete conductivity matrix Muσ is a symmetric pos-

itive semi-definite diagonal matrix and Muε is a symmetric positive definite diagonal

matrix. Furthermore the reduced discrete material matrices Muζ , Mu

ξ , MuνpCu

"auq andthe reduced discrete differential reluctivity matrix Mu

ν,dpCu"auq are positive definite, see

Remark 3.33.

Property 3.32. Let Assumption 3.28 be fulfilled. We have:

Muν “ Mu

ζGuMuξrSuMu

ζ

Gu has full column rank

CuΛu has full column rank

Gu “ ´rSJu , rGu “ ´SJu and rCu “ CJu

rSu

rCu “ 0 and CuGu “ 0

In the following we suppose that Assumption 3.31 and Property 3.32 are valid.

Maxwell’s Grid Equations using Coulomb Gauge

First we focus a Coulomb gauge, that is, ϑ “ 0, and we obtain a DAE of the type

A py, tq d

dtd py, tq ` b py, tq “ 0 (3.61)

with

A “»

–

0 0I 00 I

fi

fl , d py, tq “ˆ

MuεGuφu `Mu

ε"πu

"au

˙

and

b py, tq “¨

˝

rSuMuν

"au

MuσGuφu `Ku

νp"auq"au `Muσ

"πu ` ddt

MuεGbφb `Mu

σGbφb

´"πu

˛

‚,

where Kuνp"auq “ rCuMu

νpCu"auqCu. The DAE (3.61) has a properly stated leading term.

With

D py, tq “„

MuεGu 0 Mu

ε

0 I 0

.

it is easy to verify and we can choose R “ I.

84


The next steps are: First we determine the higher index components. With that it iseasy to show that the index is always greater than one. Finally we show that the indexis always two. Following the index analysis we present an approach to compute suitablestarting values for the numerical integration.

We determine the index of the DAE (3.61). We start with the first matrix of the matrixchain, see Definition 2.21, given by

G0 py, tq “»

–

0 0 0MuεGu 0 Mu

ε

0 I 0

fi

fl .

Obviously the matrix G0 py, tq is always singular and thus the DAE (3.61) has not index-0, see Lemma 2.30. A projector onto ker G0 py, tq is given by

Q0 “»

–

I 0 00 0 0´Gu 0 0

fi

fl .

For the matrix chain we need the derivative of b py, tq with respect to the unknowns.

Remark 3.33 ([DMW08, Sch11]). The derivative of Kuνp"auq “ rCuMu

νpCu"auqCu with

respect to "au is given by

Kuν,dp"auq “ rCuMu

ν,dpCu"auqCu

with:

d

d"au

”

rCuMuνpCu

"auqCu"au

ı

“ d

d"au

”

rCuMuνp

""

buq""

bu

ı

“ rCud

d""

bu

”

Muνp

""

buq""

bu

ı d

d"au

""

bu

“ rCud

d""

bu

”

Muνp

""

buq""

bu

ı d

d"au

rCu"aus

“ rCud

d""

bu

”

Muνp

""

buq""

bu

ı

Cu

“ rCuMuν,dpCu

"auqCu

For Brauer’s model the reduced discrete differential reluctivity matrix Muν,dpCu

"auq ispositive define, see Corollary A.13. in [Sch11].

Then we get

B0 py, tq “»

–

0 rSuMuν 0

MuσGu Ku

ν,dp"auq Muσ

0 0 ´I

fi

fl

85

and

B0 py, tqQ0 “»

–

0 0 00 0 0

Gu 0 0

fi

fl .

The next step is the calculation of the intersection of N0 and S0 py, tq. That intersectionis crucial for the index and for the consistent initialization as well. The intersection ofN0 and S0 py, tq can be described as follows.

Lemma 3.34. The Assumption 3.31 and Property 3.32 holds true. The index-1 set ofthe DAE (3.61) can be described by

N0 X S0 py, tq “ im Q0

for all py, tq P D ˆ I.

Proof . For calculating the index-1 set we make use of Remark 2.27. For a suitabledescription we need a projector along im G0 py, tq. In order to determine such a projectorwe calculate a projector onto ker G0 px, tqJ, see Remark A.8, with

G0 py, tqJ “»

–

0 ´rSuMuε 0

0 0 I0 Mu

ε 0

fi

fl .

We can choose a projector onto ker G0 py, tqJ and along im G0 py, tq by

WJ0 “

»

–

I 0 00 0 00 0 0

fi

fl and W0 “»

–

I 0 00 0 00 0 0

fi

fl .

We get

W0B0 py, tqQ0 “»

–

0 0 00 0 00 0 0

fi

fl

and hence N0 X S0 py, tq “ im Q0.

It is obvious that the index-1 set N0 X S0 py, tq is always not empty, that is, the DAE(3.61) has never index-1, see Definition 2.23. But the index does not exceed two as wewill see in the next theorem.

Theorem 3.35 (index-2). Let Assumption 3.31 and Property 3.32 be fulfilled. TheDAE (3.61) has index-2.

86


Proof . At first we need

G1 py, tq “»

–

0 0 0MuεGu 0 Mu

ε

Gu I 0

fi

fl

in order to proceed the matrix chain. For the characterization of the index-2 set weintroduce a projector along im G1 py, tq, see Remark 2.28. On this we determine aprojector onto ker G1 px, tqJ, see Remark A.8. By investigating in

G1 py, tqJ “»

–

0 ´rSuMuε ´rSu

0 0 I0 Mu

ε 0

fi

fl .

We can choose a projector onto ker G1 py, tqJ and along im G1 py, tq by WJ1 “ WJ

0 andW1 “ W0. Next we take into account

P0 “»

–

0 0 00 I 0

Gu 0 I

fi

fl , B0 py, tqP0 “»

–

0 rSuMuν 0

MuσGu Ku

ν,dp"auq Muσ

´Gu 0 ´I

fi

fl ,

where P0 is the complementary projector to Q0, and

W1B0 py, tqP0 “»

–

0 rSuMuν 0

0 0 00 0 0

fi

fl .

Let be z P ker G1 py, tq X ker W1B0 py, tqP0. That is true if and only if the conditions

z"πu“ ´Guzφu (3.62)

z"au“ ´Guzφu (3.63)

rSuMuνz"au

“ 0 (3.64)

are fulfilled. Left-multiplying (3.63) by rSuMuν and using (3.64) yields

rSuMuνGuzφu “ 0

and hence zφu “ 0 due to the choice of Muν “ Mu

ζGuMuξrSuMu

ζ . From (3.62) and (3.63) we

get`

z"au, z"πu

˘ “ 0 and conclude z “ 0, see Definition 2.23.

In order to start the integration of the DAE (3.61) we need a consistent initialization.For the index-2 case we apply Theorem 2.58.

Assumption 3.36. For the DAE (3.61) exists the continuous partial derivatives B

Btd py, tq

and B

BtW1b py, tq for all py, tq P D ˆ I.

87

These assumptions are not a restriction since, if a solution exists, then B

Btd py, tq exists

and is continuous. Moreover W1b py, tq describes exactly the hidden constraints andhence B

BtW1b py, tq needs to exists and to be continuous to have a solution to the problem.

In addition the DAE (3.61) has a constant matrix A and there are the constant projectorsQ0 and W1. We need to show that the index-2 variables enter linearly only.

Lemma 3.37. Let Assumption 3.31 and Property 3.32 be fulfilled. The index-2 variablesenter the DAE (3.61) linearly only.

Proof . From Lemma 3.34 we easily obtain a constant projector T onto N0 X S0 py, tqgiven by Q0 and U “ P0. The unknowns are divided into

y “ Ty ` Uy “¨

˝

φu

0´Guφu

˛

‚`¨

˝

0"au

Guφu ` "πu

˛

‚.

Now we can write b py, tq “ b pUy, tq ` BTy with

B “»

–

0 0 00 0 00 0 ´I

fi

fl .

The relation d py, tq “ d pUy, tq is obvious by Lemma 2.54.

The DAE (3.61) fulfills all requirements to apply Theorem 2.58 in case of index-2. Butwe still need an operating point when we want to integrate it numerically. Since Theo-rem 2.58 is applicable to the DAE an operating point is sufficient to start the numericalintegration, see Lemma 2.61.

Lemma 3.38. Let the DAE (3.61) be given and t0 P I. An operating point pz0, y0, t0qwith z0 “

´

z0"au, z0

"πu

¯

and y0 “ pφ0u,

"a0u,

"π0uq can be calculated as follows:

Choose φ0u P Rnφ and "π0

u P Rnπ arbitrarily, and "a0u P kerrSuMu

ν .

Compute the missing parts by:

Muεz

0"au“ Mu

σGuφ0u `Ku

νp"a0uq"a0

u `Muσ

"π0u ` d

dtMuεGbφb pt0q `Mu

σGbφb pt0qz0

"πu“ "π0

u

Remark 3.39. Due to the structure of the DAE (3.61) we obtain a locally unique solu-tion through every consistent initial value and perturbation index-2, see Theorem 2.46,and 2.50.

88


Maxwell’s Grid Equations using Lorenz Gauge

Next we take Lorenz gauge into account, that is ϑ “ 1. Let Assumption 3.31 andProperty 3.32 be fulfilled. Then we obtain an ODE of the form

Ad

dty ` b py, tq “ 0 (3.65)

with

A “»

–

rSuMuεGu 0 0

MuεGu 0 Mu

ε

0 I 0

fi

fl

and

b py, tq “¨

˝

rSuMuν

"au

MuσGuφu `Ku

νp"auq"au `Muσ

"πu ` ddt

MuεGbφb `Mu

σGbφb

´"πu

˛

‚.

Lemma 3.40. Let Assumption 3.31 and Property 3.32 be fulfilled. Then, MGE (3.60)using Lorenz gauge is an ODE of the form (3.65).

Proof . From Assumption 3.31 and Property 3.32 we deduce that rSuMuεGu and Mu

ε arenonsingular. Thus,

A “»

–

rSuMuεGu 0 0

MuεGu 0 Mu

ε

0 I 0

fi

fl

is nonsingular.

Hence for Lorenz gauge we have no restriction for initial values.

Remark 3.41. The chosen gauge condition for MGE (3.60) has a huge impact on thestructure of the resulting system. In case of the Coulomb gauge we obtain an index-2DAE and for Lorenz gauge we attain an ODE. That is, from the numerical point of viewLorenz gauge is to be prefer. Next we consider the Jacobians results from integrating theDAE in time by BDF methods with step size h ą 0. For the DAE (2.30) the Jacobianreads

J py, tq “ α0

hA ptqD py, tq ` B0 py, tq .

Depending on the choice of Muζ and Mu

ξ , the MGE (3.60) using Lorenz gauge may leadsto more dense Jacobians than using Coulomb gauge. In addition the structure of theJacobians depending on the gauge. For Lorenz gauge the Jacobian reads

JL py, tq “»

–

α0

hrSuMu

εGurSuMu

ν 0`

Muσ ` α0

hMuε

˘

Gu Kuν,dp"auq

`

Muσ ` α0

hMuε

˘

0 α0

hI ´I

fi

fl

89

with a nonzero diagonal and for Coulomb gauge

JC py, tq “»

–

0 rSuMuν 0

`

Muσ ` α0

hMuε

˘

Gu Kuν,dp"auq

`

Muσ ` α0

hMuε

˘

0 α0

hI ´I

fi

fl .

Hence the Lorenz gauge system could be suitable for iterative solvers, particularly withregard to the possible large number of unknowns. Note that for sufficient small h ą 0the Jacobian JL py, tq and JC py, tq are nonsingular due to Lemma 2.6.

Remark 3.42. Coulomb gauge could be suitable for iterative solvers, too. For this weneed to reformulate the DAE (3.61). We add the Coulomb gauge (3.56), ϑ “ 0, by agrad-div formulation directly to the reduced discrete Maxwell-Ampere’s law (3.51) andwe add the reduced discrete continuity equation (3.54) to the system equations. Thisleads to the DAE

rSuMuεGu

d

dtφu ` rSuMu

ε

d

dt"πu ` rSuMu

σGuφu ` rSuMuσ

"πu

´rSuMuεΛu

d

dtvE ´ rSuMu

σΛuvE “ 0

MuεGu

d

dtφu `Mu

ε

d

dt"πu `Mu


"auqCu"au `Mu

ζGuMuξrSuMu

ζ"au `Mu

σ"πu

´MuεΛu

d

dtvE ´Mu

σΛuvE “ 0

d

dt"au ´ "πu “ 0

where the Coulomb gauge is implicitly fulfilled. The BDF Jacobian for this DAE is givenby

JC py, tq “»

–

rSu

`

Muσ ` α0

hMuε

˘

Gu 0 rSu

`

Muσ ` α0

hMuε

˘

`

Muσ ` α0

hMuε

˘

Gu Kuν,dp"auq `Mu

ζGuMuξrSuMu

ζ

`

Muσ ` α0

hMuε

˘

0 α0

hI ´I

fi

fl

with a nonzero diagonal.

Remark 3.43. It seems that MGE (3.60) using Coulomb gauge has some disadvantagescompared to Lorenz gauge. However, a reformulation of MGE (3.60) using Coulombgauge with Mu

ζ “ Muε proposed by [Jan12b] lead to an ODE if we disregard the current

equation and taking into account that rSuMuζGu and Mu

ξ are nonsingular. The idea is toexploit the kernel of the Coulomb gauge. Let tb1, . . . , bku be an orthonormal basis with

respect to the standard scalar product on Rk of kerrSuMuε . Moreover let tbk`1, . . . , bnau

be an orthonormal extension of tb1, . . . , bku to an orthonormal basis with respect to thestandard scalar product on Rna . We define BP “

“

bk`1 . . . bna

‰ P Rnaˆna´k. Then

P “ BPBJP is a projector along kerrSuMuε and we obtain

0 “ rSuMuε

"au “ rSuMuεP

"au “ rSuMuεBPBJP

"au.

90


Due to the choice of BP we obtain that rSuMuεBP is nonsingular and hence

BJP"au “ 0.

With Q “ BQBJQ, BQ ““

b1 . . . bk‰ P Rnˆk, and "aq “ BJQ

"au, "πq “ BJQ"πu we obtain

MuεGu

d

dtφu `Mu

εBQd

dt"πq `Mu

σGuφu `KuνpBQ

"aqqBQ"aq `Mu

σBQ"πq

´MuεΛu

d

dtvE ´Mu

σΛuvE “ 0

d

dt"aq ´ "πq “ 0

(3.66)

from MGE (3.60). We split the first equation of (3.66) using BJQ and BJP. We achieve

BJQMuεBQ

d

dt"πq ` BJQMu

σGuφu ` BJQKuνpBQ

"aqqBQ"aq ` BJQMu

σBQ"πq

´BJQMuεΛu

d

dtvE ´ BJQMu

σΛuvE “ 0

BJPMuεGu

d

dtφu ` BJPMu

εBQd

dt"πq ` BJPMu

σGuφu ` BJPKuνpBQ

"aqqBQ"aq ` BJPMu

σBQ"πq

´BJPMuεΛu

d

dtvE ´ BJPMu

σΛuvE “ 0

d

dt"aq ´ "πq “ 0

using Property 3.32. Since BJQMuεBQ and BJPMu

εGu are nonsingular we obtain an ODEfor pφu,

"aq,"πqq. In fact, that is some kind of index reduction using knowledge of the

solution of "au given by the Coulomb gauge.

3.3 Summary

This chapter has briefly introduced Maxwell’s equations and the finite integration tech-nique for the resulting spatial discretization. We discussed a potential formulation ofMaxwell’s equations and presented a general class of gauge conditions. Next we moti-vated Dirichlet boundary conditions for the potentials.General properties of the discrete operators in terms of the finite integration techniquewere discussed. The Maxwell’s grid equations (3.60) were formulated in terms of poten-tials with incorporated boundary conditions using a new class of discrete gauge condi-tions (3.55) in terms of the finite integration technique. We defined a suitable boundaryexcitation and formulated current equations (3.58) for the currents through the elec-tromagnetic devices to be easily accessible for circuit simulation. The chosen approachdiffers substantially from [DHW04, Ben06, BBS11, Sch11], where the excitation is con-structed using several conductor models and applied as a source term.The structural properties of Maxwell’s grid equations (3.60) formulated as a differential-algebraic equation with a properly stated leading term were discussed and analyzed by

91

the index concept to obtain new index results. The first new result was that the indexdepends on the chosen gauge condition. The Coulomb gauge leads to the locally uniquesolvable index-2 and perturbation index-2 differential-algebraic equation (3.61) formu-lated with a properly stated leading term (Theorem 3.35 and Remark 3.39) with linearindex-2 variables (Lemma 3.37) and we provided a way to calculate an operating point(Lemma 3.38) to determine a consistent initialization. Maxwell’s grid equations turnedout to be an ordinary differential equation (3.65) using Lorenz gauge (Lemma 3.40).These results were obtained without taking the currents through the device into ac-count.We analyzed the structural differences of Maxwell’s grid equations using Coulomb andLorenz gauge (Remark 3.41 and 3.42). Finally, we reformulated Maxwell’s grid equationsusing a particular Coulomb gauge as an ordinary differential equation (3.66) by anorthonormal basis decomposition (Remark 3.43) without taking the currents throughthe device into account. From the results of both ordinary differential equations it canbe concluded that the modeling of the Maxwell’s grid equations has an impact on theperturbation sensitivity and thus careful modeling is desirable.

92

4 Electric Network

Today electric networks and circuits are indispensable. They can be found in almostevery electronic device from radios to central processing units of our personal computerto smartphones. An electric network is the interconnection of elements such as con-densers, resistors, coils and batteries modeled by capacitors, resistors, inductors, currentsources and voltage sources or more complex elements such as diodes and metal-oxide-semiconductor field-effect transistor.

To reduce cost and development cycles of new electric products numerical simulationsare used to predict the circuit’s behavior in terms of physical quantities such as voltagesand currents. A suitable model for numerical simulation of electric networks has tomeet two contradicting requirements. On the one hand the physical behavior of anelectric network should be as correct as possible. On the other hand the model hasto be simple enough to keep the simulation time reasonably small. With regard tothe simulation time usually the first step is to restrict the circuit elements to the basicelements capacitors, resistors, inductors, current sources and voltage sources while otherelements are replaced by equivalent circuits, that is, basic elements only.

A well-established modeling approach to meet the requirements is the modified nodalanalysis providing a system with a relatively small dimension that is able to automat-ically setup the network equations, see [CL75, CDK87, DK84]. This model analysis issuccessfully applied in established programs such as SPICE (Electronics Research Lab-oratory of the University of California, Berkeley) and TITAN (Infineon TechnologiesAG).

For today’s challenges the circuit industry is continuously developing new circuits andcircuit elements. In 2008 HP Labs announced the physical realization of a new circuitelement, namely, the memristor, whose existence was postulated in 1971 by Leon Chua,see [Chu71, SSSW08]. This has motivated further research on memristors since manypotential applications are reported such as storing huge amount of data or replacingtransistors. The use of memristors in circuit simulation requires some effort and thememristor needs to be embedded in actual circuit models. The nodal analysis methodhas already been extended by memristor models. The index of the resulting differential-algebraic equation is investigated in [Ria10]. In this thesis we extend the modifiednodal analysis by memristor models and investigate the structural properties of resultingdifferential-algebraic equation formulated with a properly stated leading term.

This chapter is organized as follows. First, we introduce the characteristic equations

93

and topology for the basic circuit elements known from literature, [CDK87, DK84],and, in addition, for the memristor, [Chu71]. Next, we familiarize with the modifiednodal analysis. Finally we extend the modified nodal analysis by memristor modelsand the resulting differential-algebraic equation with a properly stated leading term isanalyzed in terms of the index. We extend the well-known topological index conditionsof [Tis99, ET00] for the modified nodal analysis to circuits including memristors. Inaddition, we present an approach to calculate a consistent initialization for the modifiednodal analysis including memristor models.

4.1 Network Modeling

ME are also applicable to circuits. However, the complexity of integrated circuits makessimplifications unavoidable. Therefore an independent theory was deduced from ME,tailored for circuit simulations, see [CL75].The spatial dimensions of the elements are disregarded in this investigations. Twopreconditions must be met: The electrical connections between the circuit elementshave to be ideally conducting and the nodes have to be ideal and concentrated. Thephysical behavior of the circuit elements is modeled by characteristic equations.

In the modified nodal analysis (MNA - Modified Nodal Analysis) the circuit is modeledby a network graph and the topology can be described by Kirchhoff’s laws, see [DK84,Ria08]. We restrict ourselves to elements with two contacts and terminals, respectively,that is, every circuit element is represented by an edge with a different front and backnode.

4.1.1 Basic Electric Elements

The physical behavior of each network element is modeled by the relation between itsedge currents and its edge voltages.We specify the characteristic equations for the basic elements, that is for capacitors, re-sistors, inductors, voltage and current sources, in terms of currents and voltages throughthe elements, see [CL75, CDK87]. A part from the sources characteristic equations arededucible from ME by neglecting certain effects. Capacitors store energy in their electric

current source

resistor

node

capacitor

voltage source

inductor

mass node

Figure 4.1: Symbols of circuit elements.

94

4 Electric Network

field. The electric charges of the capacitor are modeled by a function qC : RnCˆI Ñ RnC

and the characteristic equations are given by

jC “ d

dtqC pvC, tq ,

where jC, vC : I Ñ RnC are the capacitors currents, voltages and nC P N is the numberof capacitors and I Ă R.Resistors limit the flow of electrical current by generating voltage drops and may bedescribed by a function gR : RnR ˆ I Ñ RnR given by

jR “ gR pvR, tq ,where jR, vR : I Ñ RnR are the resistor currents and voltages and nR P N is the numberof resistors.Inductors store energy in their magnetic field. The magnetic flux of the inductors ismodeled by the function φL : RnL ˆ I Ñ RnL and the characteristic equation is given by

vL “ d

dtφL pjL, tq

with jL, vL : I Ñ RnL being the inductor currents and voltages and nL P N the numberof inductors.We confine our investigation to independent sources. Voltage and current sources aredistinguished by the fact that the voltage and the current are given by

vV “ vs ptq and jI “ is ptqwith vV : I Ñ RnV and jI : I Ñ RnI , where nV, nI P N is the number of voltage andcurrent sources.

4.1.2 Memristors

If it’s pinched, it’s a memristor.

Leon Chua about the characterization ofa resistance memory device, [Chu11].

In 1971 Leon Chua introduced a new circuit element named memristor [Chu71]. Hemotivated the plausibility that such a device might someday be discovered by ME, see[Chu71, AASE`10]. This element provides a nonlinear relationship between the chargeand the flux and hence it completes the conceptual symmetry with the resistor, whosecharacteristic relate current and voltage, the inductor, involving current and flux, and thecapacitor, which relates voltage and charge. In 2008, a physical model of a two-contactdevice behaving like a memristor was announced in [SSSW08]. This has motivated alot of research on this topic, and the memristor and related devices are likely to have agreat impact on electronics in the near future at the nanometer scale, see references in

95

memristor

Figure 4.2: Symbols of memristor elements.

[Ria10, Ria11]. For this reason the memristor needs to be embedded in actual circuitmodels.

Memristors are governed by charge-flux relations gM : RnMˆRnMˆI Ñ RnM of the type

gM pφ, qM, tq “ 0

with φ, qM : I Ñ RnM , nM P N is the number of memristors and I Ă R. In the followingwe assume that the devices have two contacts and are either charge-controlled, that is,the fluxes can be expressed by

φ “ φM pqM, tq ,where φM : RnM ˆ I Ñ RnM . We assume that the partial derivatives

M pq, tq “ BBqφM pq, tq

exist and is continuous. We call M : RnM ˆ I Ñ RnMˆnM the memristance. Togetherwith the basic relations

d

dtφM pqM, tq “ vM and

d

dtqM “ jM, (4.1)

where jM, vM : I Ñ RnM are the memristors currents and voltages, we can conclude

vM “ M pqM, tq jM

and it becomes clear why that devices are called memristors. In case of a constantmemristance the memristors do not differ from resistors. For a non-constant memristancethe memristance depends on

qM ptq “ż t

´8

jM pτq dτ

and hence the memristors have an memory effect.

In [Ria10, RT11, Ria11] an extension of the nodal analysis and in [BT10, FY10] anextension of the MNA are presented including memristor models. There is a number ofSPICE implementations of the memristor, see [BBB09a, KKS10, BBBK10, AASE`10]and references therein. Most SPICE models of the memristor are developed on thebasis of the HP memristor or using subcircuits to model the memristor’s behavior. In[SSSW08, KKS10, BBBK10, Chu11] memristances are given.A lot of recent research is focused on devices closely related to the memristor, such asthe memcapacitors and meminductors recently introduced in [CPD09, BBB09b]. Theseand other related circuit elements are beyond the scope of the thesis.

96

4 Electric Network

4.1.3 Network Topology and Kirchhoff Laws

We model a circuit by a directed graph G :“ pN , Eq with arbitrarily orientation, see[DK84, Ria08]. Then the network topology for elements with two contacts is retainedby the (reduced) incidence matrix A P t´1, 0, 1unNˆnb , see Appendix B. The matrix Adescribes in an elegant way the relation between all nN `1 “ |N | nodes and all ne “ |E |edges of the circuit. The (reduced) incidence is defined by:

pAqij “

$

’

&

’

%

1 if the edge j leaves node i,

´1 if the edge j enters node i,

0 else.

The reference node is called mass node and is an arbitrarily node of G.

A milestone for circuit modeling are the Kirchhoff’s laws, which deal with the conserva-tion of charge and energy in electrical circuits and were first described in 1845 by GustavKirchhoff. Both laws can be directly derived from ME, but Kirchhoff preceded Maxwelland instead generalized the work by Georg Ohm.

The Kirchhoff’s laws take into account the circuit’s topology:

(i) Kirchhoff’s voltage law (KVL - Kirchhoff’s Voltage Law): At every instant of timethe algebraic sum of voltages along each loop of the network is equal to zero.

(ii) Kirchhoff’s current law (KCL - Kirchhoff’s Current Law): At every instant of timethe algebraic sum of currents entering one node of the network is equal to zero.

KVL and KCL can be deduced from ME. We start from the following Assumptions:First, cross talk, that is, undesired capacitive, inductive, or conductive coupling fromone circuit element to another, can be neglected. Second, there is no time evolution ofthe EM fields. Last, the electrical connections between the circuit elements to be ideallyconducting and the nodes to be ideal and concentrated. If these assumptions are metME imply the Kirchhoff’s laws. KCL can be derived by the continuity equation (3.5)and KVL by Maxwell-Faraday law (3.3), respectively. In the static case ME leads to:

∇ ¨ ~Jc “ 0 (4.2)

∇ˆ ~E “ 0 (4.3)

Applying Gauss’ law to (4.2) we achieveż

F

~Jc ¨ ~nKdF “ż

V

∇ ¨ ~JcdV “ 0,

where V donates the volume and F “ BV the surface of an idealized electrical node. Thecurrent is defined by

i “ż

F

~JcdF.

97

Considering one node with edge currents i1, . . . , im with F “ řmi“1 Fi, see Figure 4.3(a),

entering this node we may describe KCL as

i2i3

im i1

(a) Node with m con-ducting edges

v2

vm

v1

(b) Loop with m con-ducting edges

Figure 4.3: KCL and KVL.

mÿ

k“1

ik “mÿ

k“1

ż

Fk

~Jc ¨ ~nKdF “ż

F

~Jc ¨ ~nKdF “ 0

that is, the sum of all edge currents entering a node equals zero. Applying Stokes’ lawon (4.3) we achieve

ż

F

~E ¨ ~nqdF “ż

E

∇ˆ ~E ¨ ~nKdE “ 0

with E “ BF and F being a loop of idealized electrical wires. The voltage is defined by

v “ż

E

~E ¨ ~nqdE

If we consider a loop with the edge voltages v1, . . . , vm with E “ řmi“1 Ei, see Fig-

ure 4.3(b), then we can formulate KVL as

mÿ

k“1

vk “mÿ

k“1

ż

Ek

~E ¨ ~nqdE “ż

E

~E ¨ ~nqdE “ 0

that is, the sum of all edge voltages in a loop equals zero.Let a connected electric network be given and j, v P Rnb be the vectors of all edge currentsand voltages. Then KCL and KVL imply

Aj “ 0 (4.4)

and

v “ AJe, (4.5)

98

4 Electric Network

where e P RnN are called node potentials, see [DK84], where the node potentials aredefined as voltage drop with respect to the mass node. The node potentials leads to asmaller system size compared to a system with the edge voltages as variables. This isdue to the fact that the network graph usually contains considerably more edges thannodes.

4.2 Modified Nodal Analysis for Circuits includingMemristors

In this section we extend the charge oriented MNA, [FG99, ET00, Gun01], for circuitsincluding memristors. We arrive at the system as [FY10], but in contrast to [FY10] weprovide a detailed analysis of the resulting DAE.

The four essential steps in setting up the equations of the MNA equations are:

(i) Apply KCL to every node, except for the mass node, that is, start from (4.4).

(ii) Replace the characteristic equations for currents of resistors, capacitors and currentsources in KCL.

(iii) Add the characteristic equations for inductors.

(iv) Add the characteristic equations for voltage sources and apply KVL (4.5) to obtaina formulation in node potentials instead of branch voltages.

The first step to gain structure information is sorting the network edges in such a waythat the incidence matrix A forms a block matrix with blocks describing the differenttypes of network elements, that is,

A “ “

AC AR AL AV AI

‰

,

where the index stands for capacitive, resistive, inductive, voltage source and currentsource edges, respectively, see [Tis99, ET00].

We are back to the MNA equations, which results in a DAE system of the form

ACd

dtqC

`

AJCe, t˘` ARgR

`

AJRe, t˘` ALjL ` AVjV ` AIis ptq “ 0

d

dtφL pjL, tq ´ AJL e “ 0

AJVe´ vs ptq “ 0

(4.6)

in time t P I, I “ rt0, T s Ă R. Denoting the number of nodes - except for the massnode - by nN , the number of inductive edges by nL and the number of voltage sourceedges by nV. The dimension of the system is nN ` nL ` nV. The given vector functions

99

qC pv, tq, gR pv, tq, φL pj, tq, vs ptq and is ptq describe the characteristic equations for thecircuit elements.The unknowns are the node potentials e : I Ñ RnN , except of the mass node, as wellas the currents jL : I Ñ RnL through inductors and the currents jV : I Ñ RnV throughvoltage sources. The potential at the mass node is assigned to zero.The first equation of (4.6) states KCL and the second one states the characteristicequations for inductances. The last equation combines the characteristic equations andKVL for the voltage sources. Details can be found in [ET00, Tis04].

Remark 4.1. In stating the model as we do we implicitly assume independent voltageand current sources only. For results with a broad class of controlled sources we refer to[ET00].

For the MNA (4.6) there are well-known index results depending on the circuit topologyonly, [Tis99, ET00]. For this we need the following assumptions and definitions.

Assumption 4.2 (no short circuit). The matrices AV and“

AC AR AL AV

‰Jhave

full column rank, that is, it exists neither a loop containing only voltage sources ( V-loop)nor a cutset containing only current sources ( I-cutset), see Remark B.14.

These assumptions are necessary for a consistent model description and very naturalsince a violation would in reality result to a short circuit. From the mathematical pointof view, the circuit equations would have either no solution or infinite many solutionsdue to KCL and KVL.

Example 4.3. The linear circuit in Figure 4.4(a) has a V-loop and the MNA (4.6) leadto

j1V ` j2V `Ge “ 0

e “ v1s ptq

e “ v2s ptq

with infinitely many solutions if and only if v1s ptq “ v2

s ptq, otherwise no solutions exist.The linear circuit in Figure 4.4(b) has an I-cutset and the MNA equations (4.6) lead to

v2s ptqe

v1s ptq

(a) V-loop

i2s ptq e2

e1

i1s ptq(b) I-cutset

Figure 4.4: Example of a V-loop and an I-cutset.

100

4 Electric Network

i1s ptq ´ i2s ptq “ 0

i1s ptq `Ge1 “ 0

with infinitely many solutions if and only if i1s ptq “ i2s ptq since e2 can be chosen freely,otherwise no solutions exists.

Assumption 4.4 (Passivity, [Bar04]). The functions qC pu, tq , φL pj, tq and gR pu, tq arecontinuously differentiable with

C pu, tq “ BBuqC pu, tq , L pj, tq “ B

BjφL pj, tq , G pu, tq “ BBugR pu, tq ,

being positive definite. Physically, we say that the elements are locally passive, that is,they do not produce any energy.

For the later analysis special loops and cutsets play a key role, [ET00].

Definition 4.5 (LI-cutset). A cutset is called LI-cutset if and only if the cutset containsonly inductors and current sources.

Definition 4.6 (CV-loop). A loop is called CV-loop if and only if the loop containsonly capacitors and voltage sources.

Theorem 4.7. Let Assumption 4.2 and Assumption 4.4 be fulfilled. The MNA (4.6)represent a DAE (2.18) with a properly stated leading term. The DAE has

index-0 if and only if there are no voltage sources in the circuit and the circuit hasa tree containing capacitors only,

index-1 if and only if there is at least a voltage sources in the circuit or there is notree containing capacitors only and if there is neither an LI-cutset nor a CV-loopwith at least one voltage source,

otherwise, it has index-2.

Proof . For the properly stated leading term we refer to [Mar03]. The index result canbe found in [Tis99] and in Theorem 4.3. in [ET00].

If the DAE (4.6) has index-2 then the numerically unstable index-2 components are givenby currents through voltage sources of CV-loops but also by potentials of inductors andcurrent sources of LI-cutsets, see [Est00, EFM`03]. Fortunately the index-2 variablesappear linearly only, see [Est00]. Using perturbation index analysis it has been shownfor index-2 Hessenberg systems with linear index-2 variables, [ASW95], and for index-2circuits, [Tis01], that the numerical difficulties in time integration are moderate, becausethe differential (index-0) variables are not affected by numerical differentiations.

Next, we add memristor elements to our system. That is, we enlarge our list of basicelements by the memristor. In the MNA framework, we simply add the unknown current

101

jM P RnM through the memristors to KCL using the corresponding incidence matrix AM

with nM the number of memristors. In addition we have to add the characteristicequations for memristors. Starting from the MNA (4.1), we obtain the extended MNAsystem

ACd

dtqC

`

AJCe, t˘` AM

d

dtqM ` ARgR

`

AJRe, t˘` ALjL ` AVjV ` AIis ptq “ 0

d

dtφM pqM, tq ´ AJMe “ 0

d


AJVe´ vs ptq “ 0

(4.7)

with the additional unknowns qM : I Ñ RnM and characteristic equations φM pq, tq.

For our later investigations we need the following assumptions.


AC AR AM AL AV

‰J

have full column rank, that is, it exists neither a V-loop nor an I-cutset, see Remark B.14.

Assumption 4.9 (Passivity). The function φM pq, tq is continuously differentiable with

M pq, tq “ BBqφM pq, tq

being positive definite.

4.3 Numerical Analysis

In this section we investigate the extended MNA system (4.7) and extend the topologicalindex results for the MNA equations (4.6). The index still depends on simple topologicalcriteria and we see that memristors behave like resistors from the index point of view.Furthermore we provide an approach to calculate a consistent initialization.

The steps are as follows: At first we show that the resulting DAE has a properly statedleading term. Then we develop network topological index-0 conditions. Next we deter-mine the higher index components. With this it is easy to formulate network topologicalindex-1 conditions. Finally we show that the index is always lower or equal two. Afterthe index analysis, we present an approach to compute suitable starting values for thenumerical integration.

We suppose that Assumption 4.4 and 4.9 are valid. The extended MNA (4.7) can bewritten as a DAE given by

Ad

dtd py, tq ` b py, tq “ 0 (4.8)

102

4 Electric Network

with unknowns y “ pe, qM, jL, jVq and the describing matrix and functions

A “

»

—

—

–

AC AM 0 00 0 I 00 0 0 I0 0 0 0

fi

ffi

ffi

fl

, d py, tq “

¨

˚

˚

˝

qC

`

AJCe, t˘

qM

φM pqM, tqφL pjL, tq

˛

‹

‹

‚

as well as

b py, tq “

¨

˚

˚

˝

ARgR

`

AJRe, t˘` ALjL ` AVjV ` AIis ptq

´AJMe´AJL e

AJVe´ vs ptq

˛

‹

‹

‚

.

Remark 4.10. In practice, a reformulation of the DAE

Ad

dtd py, tq ` b py, tq “ 0

to a DAE

Ad

dtrd py, tq ` b py, tq “ 0

with a properly stated leading term and relations

Ad

dtd py, tq “ A

d

dtrd py, tq ,rd py, tq “ rPd py, tq and ker A “ ker rP

is not necessary since we are allowed to move the constant projector rP from outside intothe time derivative and vice versa, see [Mar03]. Moreover with AD py, tq “ A B

Byrd py, tq

the matrix chain is uneffected by the reformulation, too.

Lemma 4.11. Let the Assumption 4.4 and 4.9 be satisfied. Then, the DAE (4.8) has aproperly stated leading term, where the constant projector

R “

»

—

—

–

“

AC AM

‰` “

AC AM

‰ 00

00

0 0 I 00 0 0 I

fi

ffi

ffi

fl

realizes the decomposition (2.9).

Proof . The first step is to rewrite the DAE (4.8). For that we choose the projector

R with ker A “ ker R, see Lemma A.13, where“

AC AM

‰`denote the Moore-Penrose

inverse of“

AC AM

‰

. With that we get

rd px, tq “ Rd px, tq

103

and A ddt

d py, tq “ A ddtrd py, tq holds true. We to show im R “ im B

Byrd py, tq, see Defini-

tion 2.11, that is, it remains to prove

im“

AC AM

‰` “

AC AM

‰ “ im“

AC AM

‰` “

ACC`

AJCe, t˘

AJC AM

‰

That is true since im ACC`

AJCe, t˘

AJC “ im AC, see Lemma A.3.

Remark 4.12. Note, the constant projector

rP “

»

—

—

–

A`CAC 0 0 00 A`MAM 0 00 0 I 00 0 0 I

fi

ffi

ffi

fl

does not provide a properly stated leading term since ker A Ć ker rP.

Next we determine the index of the DAE (4.8) by simple topological criteria. We startwith the first matrix of the matrix chain, see Definition 2.21, given by

G0 py, tq “

»

—

—

–

ACC`

AJC, t˘

AJC AM 0 00 M pqM, tq 0 00 0 L pjL, tq 00 0 0 0

fi

ffi

ffi

fl

(4.9)

with

D py, tq “

»

—

—

–

C`

AJCe, t˘

AJC 0 0 00 I 0 00 M pqM, tq 0 00 0 L pjL, tq 0

fi

ffi

ffi

fl

.

If the matrix G0 py, tq is nonsingular, all equations are differential equations, such thatthe problem is an ODE. This is the case for the following class of circuits.

Theorem 4.13 (index-0). Suppose Assumption 4.4 and 4.9 hold true. The DAE (4.8)has index-0 if and only if there is a tree containing capacitors only and no voltage source.

Proof . We have to check under which conditions the matrix G0 py, tq is nonsingular.Since C

`

AJCe, t˘

, M pqM, tq and L pjL, tq are positive definite this is the case if and onlyif the zero rows and columns disappear and ker AJC “ t0u, see Lemma A.3. The nullspace of AJC is trivial if and only if the circuit has a tree containing capacitors only, seeTheorem B.11. The zero rows and columns disappear if and only if no voltage sourcesexist. Using Lemma 2.30 we can conclude that the DAE has index-0.

To further continue the matrix chain we need a projector onto ker G0 py, tq. A possiblechoice for such a projector is

Q0 “

»

—

—

–

QC 0 0 00 0 0 00 0 0 00 0 0 I

fi

ffi

ffi

fl

104

4 Electric Network

due to Assumption 4.4, where QC is constant projectors onto ker AJC. For the matrixchain we need the derivative of b py, tq with respect to the unknowns which is given by

B0 py, tq “

»

—

—

–

ARG`

AJRe, t˘

AJR 0 AL AV

ÁJM 0 0 0ÁJL 0 0 0AJV 0 0 0

fi

ffi

ffi

fl

.

In addition we calculate

B0 py, tqQ0 “

»

—

—

–

ARG`

AJRe, t˘

AJRQC 0 0 AV

ÁJMQC 0 0 0ÁJL QC 0 0 0AJVQC 0 0 0

fi

ffi

ffi

fl

.

As already mentioned with regard to the analysis, certain loops and cutsets of edgesplay a key role. In order to describe different circuit configurations in more detail wewill introduce some useful projectors. We denote by

QC´V and QCRMV

projectors onto

ker QJCAV and ker

“

AC AR AM AV

‰J

respectively, see [ET00]. The next lemmata are basically known from [ET00] and slightlyextend them to circuits including memristors.

Lemma 4.14 (LI-cutsets). Let a connected circuit be given. The circuit does notcontain an LI-cutset if and only if the projector QCRMV is equal to the zero matrix.

Proof . See Lemma C.2 with AR ““

AR AM

‰

and AV “ AV.

Lemma 4.15 (CV-loops). The circuit does not contain a CV-loop with at least onevoltage source if and only if the projector QC´V is equal to the zero matrix.

Proof . See Lemma C.4 with AV “ AV.

The next step is the calculation of N0 X S0 py, tq. This intersection is crucial for indexdetermination and the consistent initialization as well.

Lemma 4.16. Assume Assumption 4.4 and 4.9 to be satisfied. The index-1 set of theDAE (4.8) can be described by

N0 X S0 py, tq “ tz P Rn|ze P im QCRMV, zjV P im QC´V, zqM“ 0, zjL “ 0u

for all py, tq P D ˆ I.

105

Proof . For calculating the index-1 set we make use of Remark 2.27. For a suitabledescription we make use of a projector along im G0 py, tq. We are given one by

W0 py, tq “

»

—

—

–

QJC ´QJ

CAMM pqM, tq´1 0 00 0 0 00 0 0 00 0 0 I

fi

ffi

ffi

fl

,

see Lemma C.13, and we get

W0 py, tqB0 py, tqQ0 “

»

—

—

–

QJCG pe, qM, tqQC 0 0 QJ

CAV

0 0 0 00 0 0 0

AJVQC 0 0 0

fi

ffi

ffi

fl

with G pe, qM, tq “`

ARG`

AJRe, t˘

AJR ` AMM pqM, tq´1 AJM˘

.

Let be z P im Q0 X ker W0 py, tqB0 py, tqQ0. That is true if and only if

QCze “ ze (4.10)

zqM“ 0

zjL “ 0

QJC

`

ARG`

AJRe, t˘

AJR ` AMM pqM, tq´1 AJM˘

QCze `QJCAVzjV “ 0 (4.11)

AJVQCze “ 0 (4.12)

hold true, using Assumption 4.4 and 4.9. Left-multiply (4.11) by zJe and using (4.12)

leads to QCze P ker“

AR AM

‰J, see Lemma A.3. We obtain ze P im QCRMV in combina-

tion with (4.10) and (4.12). Thus (4.11) leads to

QJCAVzjV “ 0 and zjV P im QC´V.

We get z P N0 X S0 py, tq if and only if

ze P im QCRMV

zqM“ 0

zjL “ 0

zjV P im QC´V

holds true.

Remark 4.17. It is possible to choose a constant projector along im G0 py, tq, sinceim A “ im G0 py, tq, see Lemma 2.12. Nonetheless it is more convenient to make use ofthe given non-constant projector W0 py, tq to prove Lemma 4.16.

With the characterization of N0 X S0 py, tq we are able to provide network topologicalindex-1 conditions. We show that, from the index point of view, the memristors behavelike resistors.

106

4 Electric Network

Theorem 4.18 (index-1). Let Assumption 4.4 and 4.9 to be true. The DAE (4.8) hasindex-1 if and only if there is at least a voltage sources in the circuit or there is no treecontaining capacitors only and if there is neither an LI-cutset nor a CV-loop with atleast one voltage source.

Proof . We make use of the representation of N0XS0 py, tq as proposed in Lemma 4.16.The intersection N0 X S0 py, tq is trivial if and only if QCRMV “ 0 and QC´V “ 0. Thisis equivalent the condition that to the circuit containing neither LI-cutsets nor CV-loops, see Lemma 4.14 and 4.15. Using Definition 2.23 we get the DAE (4.8) to be ofindex-1.

The DAE (4.8) can be of index-2 also, but higher index problems can be avoided as wewill see in the next theorem. We will see that LI-cutsets and CV-loops are the onlycritical circuit configurations.

Obviously the dimension of N0XS0 px, tq is constant, which is important for the index-2case.

Theorem 4.19 (index-2). Let Assumption 4.4, 4.8 and 4.9 hold true. The DAE (4.8)has index-2 if and only if there is an LI-cutset or a CV-loop with at least one voltagesource.

Proof . At first we need

G1 py, tq “

»

—

—

–

ACC`

AJCe, t˘

AJC ` ARG`

AJRe, t˘

AJRQC AM 0 AV

ÁJMQC M pqM, tq 0 0ÁJL QC 0 L pjL, tq 0AJVQC 0 0 0

fi

ffi

ffi

fl

(4.13)

in order to proceed the matrix chain. For the characterization of the index-2 set weintroduce a projector along im G1 py, tq, see Remark 2.28. We are given one by

W1 “

»

—

—

–

QJCRMV 0 0 00 0 0 00 0 0 00 0 0 QJ

C´V

fi

ffi

ffi

fl

,

see Lemma C.14. Next we take into account

P0 “

»

—

—

–

PC 0 0 00 I 0 00 0 I 00 0 0 0

fi

ffi

ffi

fl

, B0 py, tqP0 “

»

—

—

–

ARG`

AJRe, t˘

AJRPC 0 AL 0ÁJMPC 0 0 0ÁJL PC 0 0 0AJVPC 0 0 0

fi

ffi

ffi

fl

,

where P0 is the complementary projector to Q0 and

W1B0 py, tqP0 “

»

—

—

–

0 0 QJCRMVAL 0

0 0 0 00 0 0 0

QJC´VAJVPC 0 0 0

fi

ffi

ffi

fl

.

107


QJCRMVALzjL “ 0 (4.14)

QJC´VAJVPCze “ 0 (4.15)

QCze P im QCRMV (4.16)

zjV P im QC´V (4.17)

zqM“ 0

PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV (4.18)

L pjL, tq´1 AJL QCze “ zjL (4.19)

are fulfilled, using

N1 py, tq “

z P Rn|QCze P im QCRMV, zjV P im QC´V, zqM“ 0,

PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV ,L pjL, tq´1 AJL QCze “ zjL

(

,

see Lemma C.15, where HC

`

AJCe, t˘ “ ACC

`

AJCe, t˘

AJC `QJCQC is positive definite, see

Lemma A.10, using Assumption 4.4 and 4.9. From (4.16) we deduce QCze “ QCRMVQCze.Left-multiplying (4.14) by zJe QJ

C and inserting of (4.19) yields

zJe QJCL pjL, tq´1 AJL QCze “ 0 and QCze P ker AJL ,

see Lemma A.3. Hence QCze P ker“

AC AV AR AM AL

‰Jand we conclude that

QCze “ 0, since I-cutsets are prohibited. Consequently (4.19) leads to zjL “ 0. Inserting(4.18) in (4.15) and using (4.17) we obtain AVzjV “ 0. Thus zjV “ 0 due to V-loops areforbidden. From (4.18) we get ze “ 0 and we result in z “ 0, see Definition 2.23.

To start the integration of the DAE (4.8) we need a consistent initialization. In case ofindex-1 we make direct use of Theorem 2.51. In case of index-2 we apply Theorem 2.58.For this we need to check the requirements.

Assumption 4.20. For the DAE (4.8) exist the continuous partial derivatives B

Btd py, tq

and B



Btd py, tq exists


BtW1b py, tq needs to exists and to be continuous to have a solution of the problem.

In addition, the DAE (4.8) has a constant matrix A and there are the constant projectorsQ0 and W1. It remains to show that the index-2 variables enter linearly only.

Lemma 4.21. The relation

gR

`

AJRe, t˘ “ gR

`

AJRPCRMVe, t˘

holds true for all pe, tq P RnN ˆ I.

108

4 Electric Network

Proof . We apply the mean value theorem. We get

gR

`

AJRe, t˘´ gR

`

AJRPCRMVe, t˘ “

ż 1

0

G`

sAJRe` p1´ sqAJRPCRMVe, t˘

AJRQCRMVeds

“ 0.

Lemma 4.22. Let Assumption 4.4 and 4.9 be fulfilled. The index-2 variables enter theDAE (4.8) linearly only.

Proof . From Lemma 4.16 we easily obtain a constant projector T onto N0 X S0 py, tqgiven by

T “

»

—

—

–

QCRMV 0 0 00 0 0 00 0 0 00 0 0 QC´V

fi

ffi

ffi

fl

and the complementary projector U reads

U “

»

—

—

–

PCRMV 0 0 00 I 0 00 0 I 00 0 0 PC´V

fi

ffi

ffi

fl

.

The unknowns are divided into

y “ Ty ` Uy “

¨

˚

˚

˝

QCRMVe00

QC´VjV

˛

‹

‹

‚

`

¨

˚

˚

˝

PCRMVeqM

jLPC´VjV

˛

‹

‹

‚

.


B “

»

—

—

–

0 0 0 AV

0 0 0 0´AJL 0 0 0

0 0 0 0

fi

ffi

ffi

fl

,

using Lemma 4.21. The relation d py, tq “ d pUy, tq is obvious by Lemma 2.54.


109

The computation of an operating point and DC (direct current) solution is very wellestablished area, see [CL75, CDK87, DK84, SV93], and is usually the first step in thesimulation of ciruits. Common approaches are, among others, homotopy methods andsource ramping, which are used in SPICE and TITAN, [Vla94, SW96, Dau10].

Let be z0 “ `

z0C, z

0M, z

0φM, z0

L

˘

, y0 “ pe0, q0M, j

0L, j

0Vq and t0 P I. We choose pz0

C, z0Mq and

pq0M, j

0Lq such that

ACz0C ` AMz0

M ` ALj0L ` AIis pt0q P im“

AR AV

‰

. (4.20)

Definition 4.23 ( RV-path). A path is called RV-path if and only if the path containingresistors and voltage sources only.

Remark 4.24. The condition (4.20) can be fulfilled if we choose all currents throughcapacitors, memristors and inductors to be zero. For capacitors, memristors and in-ductors, where the elements contacts are connected by a RV-path we can choose freelythe currents and charges through the elements, respectively. For simplicity we assumethe currents through current sources to be zero if the contacts are not connected by aRV-path, otherwise we apply a source ramping, see [Vla94].

Next we determine pe0, j0Vq. For this we need a solution of the nonlinear system:

ARgR

`

AJRe0, t0

˘` AVj0V “ ´ACz0C ´ AMz0

M ´ ALj0L ´ AIis pt0qAJVe0 “ vs pt0q

(4.21)

The Jacobian of the nonlinear system (4.21) has the form

Jev

`

e0, j0V˘ “

„

ARG`

AJRe0, t0

˘

AJR AV

AJV 0

.

Let be v P ker Jev pe0, j0Vq. Then we get:

ARG`

AJRe0, t0

˘

AJRv1 ` AVv2 “ 0 (4.22)

AJVv1 “ 0 (4.23)

Left multiplying (4.22) by vJ1 and using (4.23) leads to v1 P ker“

AR AV

‰Jand hence

ker Jev

`

e0, j0V˘ “ ker

“

AR AV

‰J ˆ t0u . (4.24)

In analogy we show ker Jev pe0, j0Vq “ ker Jev pe0, j0VqJ.

Approach 4.25. To solve the system (4.21) we suggest two possible ways:

(i) Assume“

AR AV

‰

to have full row rank. In terms of network configurations thatmeans there is a tree containing voltage sources and resistors only, that is, theindex-2 configurations are CV-loops only. That is, we can choose pz0

C, z0Mq and

110

4 Electric Network

pq0M, j

0Lq arbitrarily. Let be v P ker Jev pe0, j0Vq. Then we deduce from (4.24) the

relation v1 P ker“

AR AV

‰Jand v1 “ 0 due to

“

AR AV

‰

has full row rank.Furthermore we get v2 “ 0 by Assumption 4.8, that is, V-loops are forbidden.With that we conclude Jev pe0, j0Vq to be is nonsingular. Hence we obtain a uniquesolution for pe0, j0Vq by solving the nonlinear system (4.21) using, for example,Newton’s method.

(ii) We apply Theorem 2.59 to (4.21). Then

`

e0, j0V˘ “ BPv ` u

with u P ker BPBJP being arbitrarily and v P Rk is the unique solution of

BJPf pBPv, t0q “ 0,

where

f ppe, jVq , tq “ˆ

ARgR

`

AJRe, t˘` AVjV ` ACz0

C ` AMz0M ` ALj0L ` AIis ptq

AJVe´ vs ptq˙

.

After applying Approach 4.25 we compute the missing parts by

z0φM“ AJMe0 and z0

L “ AJL e0.

Remark 4.26. Due to the structure of the DAE (4.8) we obtain a locally unique solutionthrough every consistent inital value and the perturbation index to be not greater thantwo, see Theorem 2.33, 2.46, 2.49 and 2.50.

4.4 Summary

In this chapter we have introduced the modified nodal analysis to model circuits con-taining the basic elements including memristors formulated as a differential-algebraicequation (4.8) with a properly stated leading term.We extended the well-known topological index conditions of [Tis99, ET00] for the mod-ified nodal analysis to circuits including memristors (Theorem 4.13, 4.18 and 4.19) andshowed that the index does not exceed two. We conclude that, from index point ofview, the memristors behave like resistors. Moreover. we have shown perturbation andsolvability results for the modified nodal analysis including memristors (Remark 4.26)and the perturbation index does not exceed two.We presented two approaches (Approach 4.25) for the calculation of an operating point.Based on the linearity of the index-2 components (Lemma 4.21) the calculation of aconsistent initialization is possible by correcting an operating point by solving a linearsystem. Due to the structure it is sufficient to start the numerical integration with theimplicit Euler using an operating point to obtain a consistent initialization after the firsttime step.

111

5 Coupled ElectromagneticField/Circuit Models

Usually in a technology computer aided design environment devices exhibiting multi-physical effects such as electromagnetic or semiconductor devices are simplified and thedevices are modeled by an equivalent circuits.

The rapid developments in chip technology lead the devices being ever more minimizedand higher frequencies evoking effects that no longer can be reproduced by an equivalentcircuit in an appropriate manner. One reason is that the performance of the devices issignificantly influenced by the surrounding circuitry such as, for example, heating orinductive coupling. This requires additional iterations during the circuit design for theextraction and generation of equivalent circuit parameters for the different time stepsin simulation. Today, the equivalent circuits such as the BSIM6 transistor models (Uni-versity of California Berkeley Device Group) depend on up to hundreds of parameters.Most of these parameters do not have a direct physical interpretation, see [DF06], andtheir calibration is a time consuming and challenging task.

To meet future demands in circuit design it is recommended to combine circuit simu-lation directly with device simulation. While most elements are modeled by equivalentcircuits we simulate a particular device with a refined model to meet the contradictingrequirements of correct physical behavior of the circuit and reasonably small simulationtime.

In engineering it is a common task to couple circuit and device simulation, see [Tis04]and references therein. But for mathematics it is a young research area. Several indexresults for circuits with various distributed elements leading to differential-algebraicequations have beed proposed during the last years: lossy transmission lines [Gun01],heating [Bar04, Cul09] and semiconductors [Tis04, ST05, Sel06, Bod07, BST12]. Moretheoretical results concerning solvability of abstract differential algebraic equations, thatis, differential-algebraic equations on infinite dimensional Banach spaces, are presentedin [Tis04, Rei06, Mat12] with their results being also applicable to circuits includingpartial differential equation models.

We investigate coupled electromagnetic device/circuit models with spatially resolvedelectromagnetic devices. The electromagnetic devices are described by Maxwell’s equa-tions in a potential formulation and spatially discretized by the finite integration tech-

113

nique. Coupled magnetoquasistatic device/circuit models are investigated in [HM76,KMST93, DHW04, DW04] and index results are presented in [Tsu02, Ben06, BBS11,Sch11] using certain conductor models like stranded and solid conductors. The magne-toquasistatic assumption leads to the eddy-current problem for the device. In [Tsu02,Ben06] index-1 circuit configurations are investigated while [BBS11, Sch11] take generalcircuit configurations into account by extending the topological index conditions for themodified nodal analysis given in [Tis99, ET00]. Our index analysis for coupled electro-magnetic device/circuit models does not cover a special class of conductor models andwe do not suppose that the magnetoquasistatic assumption holds. It turns out that theindex of the coupled system depends on the chosen gauge condition. For the coupledelectromagnetic device/circuit model using Lorenz gauge we extend the topological in-dex conditions for the modified nodal analysis. The Coulomb gauge always results toan index-2 differential-algebraic equation.

This chapter is devoted to the index analysis of coupled systems. First, we introducethe terminology of coupled system and point out why an index analysis is necessary.Next, we introduce the coupled system consisting of circuits refined by spatially re-solved electromagnetic devices modeled by the modified nodal analysis and spatial dis-cretized Maxwell’s equations formulated as a differential-algebraic equation with a prop-erly stated leading term. We generalize the topological index criteria for the modifiednodal analysis to the coupled system. In addition, we present an approach to calculatea consistent initialization.

5.1 Simulation of Coupled Systems

Mathematical models for coupled systems are characterized by their decomposition intodifferent subsystems described by differential equations in space and time. These sub-systems may arise through refined modeling. The interdependencies are named couplingconditions and describe the mutal impact of the subsystems. There are two majorapproaches for the time integration of coupled systems:

cosimulation: The subsystems are solved sequentially or in parallel. The infor-mation interchange is restricted to particular time points. All subsystems maybe solved on their own time scale (multirate) with tailor-made methods (multi-method). We call cosimulation systems weakly coupled. Cosimulation requiresmore detailed analysis of the system formulation and the coupling conditions.

monolithic: All subsystems are combined into one single system of equations andsolved simultaneously. Every subsystem has all system information at every timepoint. All subsystems must be solved on the same time scale using the samemethods. We call monolithic systems strongly coupled.

In this thesis monolithic systems of DAEs are investigated. We would like to stress thatit is not sufficient to determine the index of the different subsystems to deduce the index

114

5 Coupled Electromagnetic Field/Circuit Models

of the coupled system as shown by the following examples. The index of the coupledsystem depends on the structure of the subsystems as well as on the structure of thecoupling conditions.

Example 5.1. We show that coupling of an index-1 and index-2 DAE can result in amonolithic index-1 or index-2 DAE. Let us consider the following system

d

dtx1 “ x2

d

dty1 “ y2

x2 “ u ptq y1 “ v ptqwhere u ptq and v ptq are given inputs and the subsystems consist of an index-1 and anindex-2 DAE. The coupling conditions are given by

(i)

u “ y1 ` y2 and v “ x1 ` x2

(ii)

u “ y1 and v “ x1

such that the monolithic system using (i) leads to an index-1 and using (ii) leads to anindex-2 DAE.

Example 5.2. We show that coupling two index-1 DAEs can result in a monolithicindex-2 DAE and vice versa. Let us consider the following systems

(i)

d

dtx1 “ x2

d

dty1 “ y2

x1 ` x2 “ u ptq y1 ` y2 “ v ptq

(ii)

d

dtx1 “ x2

d

dty1 “ y2

x1 “ u ptq y1 “ v ptq

where u ptq and v ptq are given inputs and the subsystem (i) consists of an index-1 and(ii) of an index-2 DAE. The coupling conditions are given by

u “ y2 and v “ x2

such that the monolithic system (i) leads to an index-2 and (ii) leads to an index-1 DAE.

115

5.2 Electromagnetic Field/Circuit Model

In this section we investigate circuits refined by spatially resolved EM devices modeledby the MNA and ME. The MNA describes the non-critical circuit parts for which amodeling using basic elements only is sufficient. Critical circuit parts for which theMNA approach is insufficient to describe the EM device behavior are modeled by MEdirectly. For simplicity we assume that only one critical EM device is given.We have to include the EM device into the MNA framework. For this charge conservationis essential which is given by the additional mass contact. We suppose that the EM devicehas nE disjoint conductive contacts and each contact of the device is joined to a nodeof the circuit. In addition we suppose that the mass contact is connected to the massnode. The contacts of the EM device joined to the same node of the electrical circuitdefine a terminal, see [Tis04, Bod07]. Let nT ` 1 be the number of terminals of the EMdevice and nN be the number of circuit’s nodes except the mass node. We define thefollowing (reduced) incidence matrix AE P t´1, 0, 1unNˆnT by

pAEqij “

$

’

&

’

%

1 if terminal j is joined to node i,

´1 if the reference terminal is attached to node i,

0 else.

The coupling of the EM device to the circuit is established by the applied node potentialsat the EM device conductive contacts and the currents through it. For this we need toadd the EM device to our list of elements. In the MNA framework we simply add thecurrent through the EM device to the KCL using the corresponding incidence matrixAE. In addition we add the MGE (3.60) for the EM device to the MNA (4.6), where theDirichlet boundary conditions for the scalar potentials are described by e “ AJEe. Thatis, we apply the potential difference to the mass node at the conductive contacts. Thatis possible since the scalar potentials are determined up to a constant. The coupled EMdevice/circuit system with gauge condition reads

ACd

dtqC

`

AJCe, t˘` ARgR

`

AJRe, t˘` ALjL ` AVjV ` AEjE ` AIis ptq “ 0

d


AJVe´ vs ptq “ 0

jE ´ ΛJu Kuνp"auq"au “ 0

ϑrSuMuεGu

d

dtφu ` rSuMu

ν"au “ 0

´MuεΛuAJE

d

dte`Mu

εGud

dtφu `Mu

ε

d

dt"πu ´Mu

σΛuAJEe

`MuσGuφu `Ku

νp"auq"au `Muσ

"πu “ 0

d

dt"au ´ "πu “ 0

(5.1)

116


in time t P I, I “ rt0, T s Ă R, see Remark 3.33.

For our analysis of the coupled system (5.1) certain loops and cutsets play a key role.


AC AR AL AE AV

‰J

have full column rank, that is, it exists neither a V-loop nor an I-cutset, see Remark B.14.

Definition 5.4 (LEI-cutset). A cutset is called LEI-cutset if and only if the cutsetcontains only inductors, EM devices and current sources.

In order to describe different circuit configurations in more detail we will introduce someuseful projectors. We denote by

QC, QC´V and QCRV

projectors onto

ker AJC, ker QJCAV and ker

“

AC AR AV

‰J

respectively, see [ET00]. The next lemmata are basically known from [ET00] and weslightly extend them to circuits including EM devices.

Lemma 5.5 (LEI-cutsets). Let a connected circuit be given. The circuit does notcontain an LEI-cutset if and only if the projector QCRV is equal to the zero matrix.

Proof . See Lemma C.2 with AR “ AR and AV “ AV.

Lemma 5.6 (CV-loops). The circuit does not contain a CV-loop with at least onevoltage source if and only if the projector QC´V is equal to the zero matrix.

Proof . See Lemma C.4 with AV “ AV.

5.3 Numerical Analysis

In this section we investigate the coupled system (5.1) using Coulomb and Lorenz gauge.For both systems we extend the topological index results for the MNA (4.6), see [Tis99,ET00]. The index depends still on simple topological criteria and we see that an EMdevice using Lorenz gauge, from the index point of view, behaves like an inductor.Furthermore we provide an approach to calculate a consistent initialization.

We suppose that Assumption 4.4, 3.31 and Property 3.32 are fulfilled.

The steps are as follows: First we show that the resulting DAEs have a properly statedleading term. Then we develop network topological index-0 conditions. Next we deter-mine the higher index components. With this it is easy to formulate network topologicalindex-1 conditions. Finally we show that the index is always lower or equal two. Afterthe index analysis, we present an approach to compute suitable starting values for thenumerical integration.

117

5.3.1 Field/Circuit System using Coulomb Gauge

The coupled system (5.1) using Coulomb gauge, that is, ϑ “ 0, can be formulated as aDAE given by

Ad

dtd py, tq ` b py, tq “ 0 (5.2)

with unknowns y “ pe, jL, jV, jE, φu,"au,

"πuq and the describing matrix and functions

A “

»

—

—

—

—

—

—

—

—

–

AC 0 0 00 I 0 00 0 0 00 0 0 00 0 0 00 0 0 Mu

ε

0 0 I 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

, d py, tq “

¨

˚

˚

˝

qC

`

AJCe, t˘

φL pjL, tq"au

´ΛuAJEe`Guφu ` "πu

˛

‹

‹

‚

as well as

b py, tq “

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

ARgR

`

AJRe, t˘` ALjL ` AVjV ` AEjE ` AIis ptq

´AJL eAJVe´ vs ptq

jE ´ ΛJu Kuνp"auq"au

rSuMuν

"au

´MuσΛuAJEe`Mu

σGuφu `Kuνp"auq"au `Mu

σ"πu

´"πu

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚

.

First, we show that the DAE has a properly stated leading term.

Lemma 5.7. Let Assumption 4.4 and 3.31 be fulfilled. Then, the DAE (5.2) has aproperly stated leading term where the constant projector

R “„

A`CAC 00 I


Proof . The first step is to rewrite the DAE (5.2). For that we choose a projector

rP “„

A`CAC 00 I

with A “ ArP, where A`C denote the Moore-Penrose inverse of AC. With this we get

rd px, tq “ rPd px, tq

118


and A ddt

d py, tq “ A ddtrd py, tq holds true. We denote rD py, tq “ B

Byrd py, tq given by

rD py, tq “

»

—

—

–

A`CACC`

AJCe, t˘

AJC 0 0 0 0 0 00 L pjL, tq 0 0 0 0 00 0 0 0 0 I 0

´ΛuAJE 0 0 0 Gu 0 I

fi

ffi

ffi

fl

We get

ker A “ ker AC ˆ t0uand

im rD py, tq “ im A`CACC`

AJCe, t˘

AJC ˆ RnL`na`nπ ,

using Assumption 4.4 and 3.31. Applying Lemma A.1, A.19 and A.21 we obtain

im A`CACC`

AJCe, t˘

AJC “ im A`CAC “ im AJC.

Hence we can choose the projector R “ rP, see Lemma A.13.

Notice that the projector rP in Lemma 5.7 is not needed for practical computations, seeRemark 4.10.

We follow the matrix chain concept, see Definition 2.21. For that we need the matrix

G0 py, tq “

»

—

—

—

—

—

—

—

—

–

ACC`

AJCe, t˘

AJC 0 0 0 0 0 00 L pjL, tq 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

´MuεΛuAJE 0 0 0 Mu

εGu 0 Muε

0 0 0 0 0 I 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

(5.3)

with

D py, tq “

»

—

—

–

C`

AJCe, t˘

AJC 0 0 0 0 0 00 L pjL, tq 0 0 0 0 00 0 0 0 0 I 0

´ΛuAJE 0 0 0 Gu 0 I

fi

ffi

ffi

fl

.

To obtain an index-0 DAE we need to check under which conditions the matrix G0 py, tq isnonsingular. If the matrix G0 py, tq is nonsingular all equations are differential equations,such that the problem is an ODE. This is the case for the following class of circuits.

Theorem 5.8 (index-0). Let Assumption 4.4 be fulfilled. The DAE (5.2) has index-0if and only if the circuit does not contain voltage sources and EM device and if there isa tree containing capacitors only.

119

Proof . Following the proof of Theorem 4.13, the remaining zero rows and columnsdisappear if and only if there are no EM device.

To further continue the matrix chain we need a projector onto ker G0 py, tq. Let bez P ker G0 py, tq. That is true if and only if

ze P im QC

zjL “ 0

z"πu“ ΛuAJEQCze ´Guzφu

z"au“ 0

hold true, due to Lemma A.3, Assumption 4.4 and 3.31. We can choose a constantprojector onto ker G0 py, tq by

Q0 “

»

—

—

—

—

—

—

—

—

–

QC 0 0 0 0 0 00 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 I 0 00 0 0 0 0 0 0

ΛuAJEQC 0 0 0 ´Gu 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

For the matrix chain we need the derivative of b py, tq with respect to the unknownswhich is given by

B0 py, tq “

»

—

—

—

—

—

—

—

—

–

ARG`

AJRe, t˘

AJR AL AV AE 0 0 0ÁJL 0 0 0 0 0 0AJV 0 0 0 0 0 00 0 0 I 0 ´ΛJu Ku

ν,dp"auq 0

0 0 0 0 0 rSuMuν 0

´MuσΛuAJE 0 0 0 Mu

σGu Kuν,dp"auq Mu

σ

0 0 0 0 0 0 Í

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and we obtain

B0 py, tqQ0 “

»

—

—

—

—

—

—

—

—

–

ARG`

AJRe, t˘

AJRQC 0 AV AE 0 0 0ÁJL QC 0 0 0 0 0 0AJVQC 0 0 0 0 0 0

0 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

´ΛuAJEQC 0 0 0 Gu 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

see Remark 3.33.

The next step is the calculation of N0 X S0 py, tq. This intersection is crucial for indexdetermination and the consistent initialization as well.

120


Lemma 5.9. Assume Assumption 4.4, 3.31 and Property 3.32 to be satisfied. Theindex-1 set of the DAE (5.2) can be described by

N0 X S0 py, tq “

z P Rn | ze P im QCRV, zjV P im QC´V,

ΛuAJEQCRVze ´Guzφu “ z"πu,`

zjL , zjE , z"au

˘ “ 0(

.

Proof . For calculating the index-1 set we make use of Remark 2.27. For this we needa projector along im G0 py, tq. We are given one by

W0 “

»

—

—

—

—

—

—

—

—

–

QJC 0 0 0 0 0 0

0 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,


W0B0 py, tqQ0 “

»

—

—

—

—

—

—

—

—

–

QJCARG

`

AJRe, t˘

AJRQC 0 QJCAV QJ

CAE 0 0 00 0 0 0 0 0 0

AJVQC 0 0 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Let be z P im Q0 X ker W0B0 py, tqQ0. That is true if and only if

ze “ QCze (5.4)

zjL “ 0

z"au“ 0

z"πu“ ΛuAJEze ´Guzφu

QJCARG

`

AJRe, t˘

AJRQCze `QJCAVzjV “ 0 (5.5)

AJVQCze “ 0 (5.6)

zjE “ 0

hold true by taking Assumption 4.4, 3.31 and Property 3.32 into account. Left-multiplyof (5.5) by zJe and utilizing (5.6) leads to QCze P ker AJR due to Lemma A.3. In combina-tion with (5.4) and (5.6) we obtain ze P im QCRV. Moreover, (5.5) yields zjV P im QC´V.With it we obtain z P N0 X S0 py, tq if and only if

ze P im QCRV

zjL “ 0

121

z"au“ 0

z"πu“ ΛuAJEQCRVze ´Guzφu

zjV P im QC´V

zjE “ 0

hold true.

It is easy to see that the index-1 set N0XS0 py, tq is always not empty, that is, the DAE(5.2) has never index-1. But the index does not exceed two as we will see in the nexttheorem.

Theorem 5.10 (index-2). Let Assumption 4.4, 5.3, 3.31 and Property 3.32 be fulfilled.The DAE (5.2) has at most index-2. It has exactly index-2 if and only if the circuit doescontain a voltage source or if an EM device or if it has not a tree containing capacitorsonly.

Proof . For the matrix chain we need

G1 py, tq “

»

—

—

—

—

—

—

—

—

–

G pe, tq 0 AV AE 0 0 0´AJL QC L pjL, tq 0 0 0 0 0AJVQC 0 0 0 0 0 0

0 0 0 I 0 0 00 0 0 0 0 0 0


εGu 0 Muε

´ΛuAJEQC 0 0 0 Gu I 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

, (5.7)

with G pe, tq “ ACC`

AJCe, t˘

AJC`ARG`

AJRe, t˘

AJRQC and we calculate the index-2 set,see Definition 2.23 and Remark 2.28. For the subspaces needed, we have to provide aprojector along im G1 py, tq. We are given one by

W1 “

»

—

—

—

—

—

—

—

—

–

QJCRV 0 0 ´QJ

CRVAE 0 0 00 0 0 0 0 0 00 0 QJ

C´V 0 0 0 00 0 0 0 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,


P0 “

»

—

—

—

—

—

—

—

—

–

PC 0 0 0 0 0 00 I 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 I 0

´ΛuAJEQC 0 0 0 Gu 0 I

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

122


where P0 is the complementary projector to Q0, and we calculate

B0 py, tqP0 “

»

—

—

—

—

—

—

—

—

–

ARG`

AJRe, t˘

AJRPC AL 0 0 0 0 0´AJL PC 0 0 0 0 0 0AJVPC 0 0 0 0 0 0

0 0 0 0 0 ´ΛJu Kuν,dp"auq 0

0 0 0 0 0 rSuMuν 0


σGu Kuν,dp"auq Mu

σ

ΛuAJEQC 0 0 0 ´Gu 0 ´I

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and

W1B0 py, tqP0 “

»

—

—

—

—

—

—

—

—

–

0 QJCRVAL 0 0 0 QJ

CRVAEΛJu Kuν,dp"auq 0

0 0 0 0 0 0 0QJ

C´VAJVPC 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 rSuMuν 0

0 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.


QCze “ QCRVze (5.8)

zjV “ QC´VzjV (5.9)

zjE “ 0


PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV (5.11)

ΛuAJEze ´Guzφu “ z"πu(5.12)

ΛuAJEQCze ´Guzφu “ z"au(5.13)

QJCRVALzjL `QJ

CRVAEΛJu Kuν,dp"auqz"au

“ 0 (5.14)


rSuMuνz"au

“ 0 (5.16)

are fulfilled, using the representation

N1 py, tq “

z P Rn | QCze P im QCRV, zjV P im QC´V,

L pjL, tq´1 AJL QCze “ zjL , PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV , zjE “ 0,

ΛuAJEze ´Guzφu “ z"πu, ΛuAJEQCze ´Guzφu “ z"au

(

,


`

AJCe, t˘ “ ACC

`

AJCe, t˘


Lemma A.10, using Assumption 4.4, 3.31 and Property 3.32. Left-multiplying of (5.11)by zJjVQJ

C´VAJV and using (5.15) leads to

zJjVQJC´VAJVHC

`

AJCe, t˘´1

AVQC´VzjV “ 0.

123

With Lemma A.3 and (5.9) we deduce

AVzjV “ 0 and zjV “ 0

since V-loops are forbidden. From (5.11) we acquire ze P im QC and together with (5.8)we get ze P im QCRV. Combining (5.10), (5.13) and (5.14) yields

QJCRVALL pjL, tq´1 AJL QCRVze `QJ

CRVAEΛJu Kuν,dp"auqΛuAJEQCRVze “ 0,

since CuGu “ 0 holds true, and we deduce ze P ker“

AL AE

‰J. Thus we come by the

condition

ze P ker“

AC AR AL AE AV

‰J.

Because I-cutsets are forbidden, we gain ze “ 0. Then the relation (5.10) leads to zjL “ 0.

Left-multiplying (5.13) by rSuMuν and using (5.16) yields

rSuMuνGuzφu “ 0

and hence zφu “ 0 due to the choice of Muν “ Mu

ζGuMuξrSuMu

ζ . From (5.12) and (5.13) we

get`

z"au, z"πu

˘ “ 0 and we conclude z “ 0, see Definition 2.23.

To start the integration of the DAE (5.2) we need a consistent initialization. For theindex-2 case we apply Theorem 2.58. For this we need to check the requirements.


Btd py, tq

and B



Btd py, tq exists


BtW1b py, tq needs to exists and to be continuous to have a solution of the problem.

In addition, the DAE (5.2) has a constant matrix A and there are the constant projectorsQ0 and W1. It remains to show that the index-2 variables enter linearly only.

Lemma 5.12. Let Assumption 4.4, 3.31 and Property 3.32 be fulfilled. The index-2variables enter the DAE (5.2) linearly only.

Proof . From Lemma 5.9 we easly obtain a constant projector T onto N0 X S0 py, tqgiven by

T “

»

—

—

—

—

—

—

—

—

–

QCRV 0 0 0 0 0 00 0 0 0 0 0 00 0 QC´V 0 0 0 00 0 0 0 0 0 00 0 0 0 I 0 00 0 0 0 0 0 0

ΛuAJEQCRV 0 0 0 ´Gu 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

124


Furthermore the complementary projector U is given by

U “

»

—

—

—

—

—

—

—

—

–

PCRV 0 0 0 0 0 00 I 0 0 0 0 00 0 PC´V 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 I 0

´ΛuAJEQCRV 0 0 0 Gu 0 I

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and the unknowns are divided into

y “ Ty ` Uy “

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

QCRVe0

QC´VjV0φu

0ΛuAJEQCRVe´Guφu

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚

`

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

PCRVejL

PC´VjVjE0

"au

´ΛuAJEQCRVe`Guφu ` "πu

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚

.


B “

»

—

—

—

—

—

—

—

—

–

0 0 AV 0 0 0 0ÁJL 0 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 Í

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

using Lemma 4.21 without memristors. The relation d py, tq “ d pUy, tq is obvious byLemma 2.54.


Let be z0 “´

z0C, z

0L, z

0"au, z0

"πu

¯

, y0 “ pe0, j0L, j0V, j

0E, φ

0u,

"a0u,

"π0uq and t0 P I. We choose

"a0u P ker Cu and pφ0

u,"π0

uq arbitrarily. Then we get

j0E “ 0

and choose z0C and j0L such that

ÁCz0C ´ ALj0L ´ AIis pt0q P im

“

AR AV

‰

,

125

see Remark 4.24 without memristors. Next we determine pe0, j0Vq. For that we need asolution of the nonlinear system:

ARgR

`

AJRe0, t0

˘` AVj0V “ ´ACz0C ´ ALj0L ´ AIis pt0q

AJVe0 “ vs pt0q(5.17)

To obtain a solution pe0, j0Vq of (5.17) we apply Approach 4.25 without memristors. Thenwe compute the missing parts by:

z0L “ AJL e0

Muεz

0"au“ Mu

σΛuAJEe0 ´MuσGuφ

0u ´Mu

σ"π0

u

z0"πu“ "π0

u

Remark 5.13. Due to the structure of the DAE (5.2) we obtain a locally unique solutionthrough every consistent initial value and the perturbation index-2, see Theorem 2.46and 2.50.

5.3.2 Field/Circuit System using Lorenz Gauge

The coupled system (5.1) using Lorenz gauge, that is, ϑ “ 1, can be formulated as aDAE given by

Ad

dtd py, tq ` b py, tq “ 0 (5.18)

with unknowns y “ pe, jL, jV, jE, φu,"au,

"πuq and the describing matrix and functions

A “

»

—

—

—

—

—

—

—

—

–

AC 0 0 0 00 I 0 0 00 0 0 0 00 0 0 0 0

0 0 rSuMuεGu 0 0

0 0 MuεGu 0 Mu

ε

0 0 0 I 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

, d py, tq “

¨

˚

˚

˚

˚

˝

qC

`

AJCe, t˘

φL pjL, tqφu"au

´ΛuAJEe` "πu

˛

‹

‹

‹

‹

‚

as well as

b py, tq “

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

ARgR

`

AJRe, t˘` ALjL ` AVjV ` AEjE ` AIis ptq

´AJL eAJVe´ vs ptq

jE ´ ΛJu Kuνp"auq"au

rSuMuν

"au

´MuσΛuAJEe`Mu

σGuφu `Kuνp"auq"au `Mu

σ"πu

´"πu

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚

.

126


Lemma 5.14. Let Assumption 4.4 and 3.31 be satisfied. Then, the DAE (5.18) has aproperly stated leading term where the constant projector

R “„

A`CAC 00 I


Proof . We follow the idea of the proof of Lemma 5.7.

The first step is an index-0 result. For this we need the matrix, see Definition 2.21,

G0 py, tq “

»

—

—

—

—

—

—

—

—

–

ACC`

AJCe, t˘

AJC 0 0 0 0 0 00 L pjL, tq 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 rSuMuεGu 0 0


εGu 0 Muε

0 0 0 0 0 I 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

(5.19)

with

D py, tq “

»

—

—

—

—

–

C`

AJCe, t˘

AJC 0 0 0 0 0 00 L pjL, tq 0 0 0 0 00 0 0 0 I 0 00 0 0 0 0 I 0

´ΛuAJE 0 0 0 0 0 I

fi

ffi

ffi

ffi

ffi

fl

.

Theorem 5.15 (index-0). Let Assumption 4.4, 3.31 and Property 3.32 be fulfilled. TheDAE (5.18) has index-0 if and only if the circuit does not contain voltage sources and ifEM device and if there is a tree containing capacitors only.

Proof . Following the proof of Theorem 4.13, the remaining zero rows and columnsdisappear if and only if there is no EM device.

The next step is to describe the intersection index-1 set. For this we compute a projectoronto ker G0 py, tq and the derivative

B0 py, tq “

»

—

—

—

—

—

—

—

—

–

ARG`

AJRe, t˘

AJR AL AV AE 0 0 0´AJL 0 0 0 0 0 0AJV 0 0 0 0 0 00 0 0 I 0 ´ΛJu Ku

ν,dp"auq 0

0 0 0 0 0 rSuMuν 0


σGu Kuν,dp"auq Mu

σ

0 0 0 0 0 0 ´I

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Let be z P ker G0 py, tq. That is true if and only if

ze P im QC

127

zjL “ 0

z"πu“ ΛuAJEQCze

zφu “ 0

z"au“ 0

due to Lemma A.3. Hence we can choose a projector onto ker G0 py, tq by

Q0 “

»

—

—

—

—

—

—

—

—

–

QC 0 0 0 0 0 00 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

ΛuAJEQC 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and we calculate

B0 py, tqQ0 “

»

—

—

—

—

—

—

—

—

–

ARG`

AJRe, t˘

AJRQC 0 AV AE 0 0 0´AJL QC 0 0 0 0 0 0AJVQC 0 0 0 0 0 0

0 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

´ΛuAJEQC 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

see Remark 3.33. Next N0 X S0 py, tq is calculated, since the intersection plays an im-portant role for the index calculation and for the consistent initialization.

Lemma 5.16. Let Assumption 4.4, 3.31 and Property 3.32 hold true. The index-1 setof the DAE (5.18) can be described by

N0 X S0 py, tq “

z P Rn | ze P im QCRV, zjV P im QC´V,

ΛuAJEQCRVze “ z"πu,`

zjL , zjE , zφu , z"au

˘ “ 0(

.

Proof . For calculating the index-1 set we make use of Remark 2.27. For that we needa projector along im G0 py, tq. We are given one by

W0 “

»

—

—

—

—

—

—

—

—

–

QJC 0 0 0 0 0 0

0 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

128



W0B0 py, tqQ0 “

»

—

—

—

—

—

—

—

—

–

QJCARG

`

AJRe, t˘

AJRQC 0 QJCAV QJ

CAE 0 0 00 0 0 0 0 0 0

AJVQC 0 0 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

The rest of the proof is entirely analog to the proof of Lemma 5.9.

With the characterization of the intersection we are able to deduce network topologicalindex-1 conditions for the coupled system DAE (5.18). The EM devices are insert intothe circuit as a kind of controlled current sources, but the analysis show that for usingLorenz gauge they, from the index point of view, behave like inductances.

Theorem 5.17 (index-1). Let Assumption 4.4, 3.31 and Property 3.32 be true. TheDAE (5.18) has index-1 if and only if there is at least a voltage sources in the circuitor there is no tree containing capacitors only and if there is neither an LEI-cutset nor aCV-loop with at least one voltage source.

Proof . We make use of the representation of N0XS0 py, tq as proposed in Lemma 5.16.The intersection N0 X S0 py, tq is trivial if and only if QCRV “ 0 and QC´V “ 0. Thisis equivalent to the circuit containing neither LEI-cutsets nor CV-loops, see Lemma 5.5and 5.6. Using Definition 2.23 we get the DAE (5.18) to be of index-1.

The DAE (5.18) can be of index-2 also, but higher index problems can be avoided.We will see that LEI-cutsets and CV-loops are the only critical circuit configurations.Obviously the dimension of N0XS0 px, tq is constant, which is important for the index-2case.

Theorem 5.18 (index-2). Let Assumption 4.4, 5.3, 3.31 and Property 3.32 be fulfilled.The DAE (5.18) has index-2 if and only if there is an LEI-cutset or a CV-loop with atleast one voltage source.

Proof . For the matrix chain we need

G1 py, tq “

»

—

—

—

—

—

—

—

—

–

G pe, tq 0 AV AE 0 0 0´AJL QC L pjL, tq 0 0 0 0 0AJVQC 0 0 0 0 0 0

0 0 0 I 0 0 0

0 0 0 0 rSuMuεGu 0 0


εGu 0 Muε

´ΛuAJEQC 0 0 0 0 I 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

, (5.20)

129


AJCe, t˘

AJCÀRG`

AJRe, t˘

AJRQC, and we calculate the index-2 set,see Definition 2.23 and Remark 2.28. For the subspaces needed, we have to provide aprojector along im G1 py, tq. We are given one by

W1 “

»

—

—

—

—

—

—

—

—

–

QJCRV 0 0 ´QJ

CRVAE 0 0 00 0 0 0 0 0 00 0 QJ

C´V 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,


P0 “

»

—

—

—

—

—

—

—

—

–

PC 0 0 0 0 0 00 I 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 I 0 00 0 0 0 0 I 0

´ΛuAJEQC 0 0 0 0 0 I

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

where P0 is the complementary projector to Q0, and we calculate

B0 py, tqP0 “

»

—

—

—

—

—

—

—

—

–

ARG`

AJRe, t˘

AJRPC AL 0 0 0 0 0ÁJL PC 0 0 0 0 0 0AJVPC 0 0 0 0 0 0

0 0 0 0 0 ´ΛJu Kuν,dp"auq 0

0 0 0 0 0 rSuMuν 0


σGu Kuν,dp"auq Mu

σ

ΛuAJEQC 0 0 0 ´Gu 0 Í

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and

W1B0 py, tqP0 “

»

—

—

—

—

—

—

—

—

–

0 QJCRVAL 0 0 0 QJ

CRVΛJu Kuν,dp"auq 0

0 0 0 0 0 0 0QJ

C´VAJVPC 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.


QCze P im QCRV (5.21)

zjV P im QC´V (5.22)

130


zjE “ 0

zφu “ 0


PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV (5.24)

ΛuAJEze “ z"πu

ΛuAJEQCze “ z"au(5.25)

QJCRVALzjL `QJ

CRVΛJu Kuν,dp"auqz"au

“ 0 (5.26)


are fulfilled, using

N1 py, tq “



`

AJCe, t˘´1

AVQC´VzjV ,

pzjE , zφuq “ 0, ΛuAJEze “ z"πu, ΛuAJEQCze “ z"au

(

,


`

AJCe, t˘ “ ACC

`

AJCe, t˘


Lemma A.10, using Assumption 4.4, 3.31 and Property 3.32. Left-multiplying of (5.24)by zJjVQJ

C´VAJV and using (5.27) leads to

zJjVQJC´VAJVHC

`

AJCe, t˘´1

AVQC´VzjV “ 0.

With Lemma A.3 and (5.22) we deduce

AVzjV “ 0 and zjV “ 0

since V-loops are forbidden. From (5.24) we acquire ze P im QC and together with (5.21)we come by ze P im QCRV. Combining (5.23), (5.25) and (5.26) yields

QJCRVALL pjL, tq´1 AJL QCRVze `QJ

CRVAEΛJu Kuν,dp"auqΛuAJEQCRVze “ 0

since CuGu “ 0 holds true, and we deduce ze P ker“

AL AE

‰J. Thus we come by the

condition

ze P ker“

AC AR AL AE AV

‰J.

Because I-cutsets are forbidden we gain ze “ 0. Then the relations (5.23), (5.12) and(5.25) leads

`

zjL , z"au, z"πu

˘ “ 0 and we conclude z “ 0, see Definition 2.23.

The topological index results for the coupled system using Lorenz gauge are also aextentsion of the topological index results of [BBS11] for coupled MQS device/circuitsystems using Lorenz gauge.

In order to start the integration of the DAE (5.18) we need a consistent initialization.In case of index-1 we make direct use of Theorem 2.51. In case of index-2 we applyTheorem 2.58. For this we need to check the requirements.

131


Btd py, tq

and B



Btd py, tq exists


BtW1b py, tq needs to exists and to be continuous in order to have a solution of

the problem.

In addition, the DAE (5.18) has a constant matrix A and there are constant projectorsQ0 and W1. It remains to show that the index-2 variables enter linearly only.

Lemma 5.20. Let Assumption 4.4, 3.31 and Property 3.32 be fulfilled. The index-2variables enter the DAE (5.18) linearly only.

Proof . From Lemma 5.16 we easily obtain a constant projector T onto N0 X S0 py, tqgiven by

T “

»

—

—

—

—

—

—

—

—

–

QCRV 0 0 0 0 0 00 0 0 0 0 0 00 0 QC´V 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

ΛuAJEQCRV 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Furthermore, the complementary projector U is given by

U “

»

—

—

—

—

—

—

—

—

–

PCRV 0 0 0 0 0 00 I 0 0 0 0 00 0 PC´V 0 0 0 00 0 0 I 0 0 00 0 0 0 I 0 00 0 0 0 0 I 0

´ΛuAJEQCRV 0 0 0 0 0 I

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and the unknowns are divided into

y “ Ty ` Uy “

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

QCRVe0

QC´VjV000

ΛuAJEQCRVe

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚

`

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

PCRVejL

PC´VjVjEφu"au

´ΛuAJEQCRVe` "πu

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚

.

132



B “

»

—

—

—

—

—

—

—

—

–

0 0 AV 0 0 0 0ÁJL 0 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 Í

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

with Lemma 4.21 without memristors. The relation d py, tq “ d pUy, tq is obvious byLemma 2.54.


Let be z0 “´

z0C, z

0L, z

0φu, z0

"au, z0

"πu

¯

, y0 “ pe0, j0L, j0V, j

0E, φ

0u,

"a0u,

"π0uq and t0 P I. We choose

"a0u P ker Cu and pφ0

u,"π0

uq arbitrarily. Then we get

j0E “ 0

and choose z0C and j0L such that

ÁCz0C ´ ALj0L ´ AIis pt0q P im

“

AR AV

‰

,

see Remark 4.24 without memristors. Next we determine pe0, j0Vq. For that we need asolution of the nonlinear system:

ARgR

`

AJRe0, t0

˘` AVj0V “ ÁCz0C ´ ALj0L ´ AIis pt0q

AJVe0 “ vs pt0q(5.28)

To obtain a solution pe0, j0Vq of (5.28) we apply Approach 4.25 without memristors. Thenwe compute the missing parts by:

z0L “ AJL e0

SuMuεGuz0

φu“ ´rSuMu

ν"a0

u

Muεz

0"au“ ´Mu

εGuz0φu`Mu

σΛuAJEe0 ´MuσGuφ

0u ´Mu

σ"π0

u

z0"πu“ "π0

u

Remark 5.21. Due to the structure of the DAE (5.18) we obtain a locally uniquesolution through every consistent initial value and the perturbation index to be notgreater than two, see Theorem 2.33, 2.46, 2.49 and 2.50.

133

5.4 Summary

In this chapter we have introduced circuits refined by spatially resolved electromagneticdevices and modeled by the modified nodal analysis and Maxwell’s grid equations. Thecoupling is realized by the applied potential at the conductive contacts of the electro-magnetic device and by the current through it. We discussed the structural propertiesof the coupled electromagnetic device/circuit system. The chosen coupling approach isdifferent to [DHW04, Ben06, BBS11, Sch11], where the coupling is realized using serveralconductor models and applied as a source term.We generalized the well-known topological index conditions of [Tis99, ET00] for themodified nodal analysis to circuits refined by spatially resolved electromagnetic devicesmodeled using Lorenz gauge (Theorem 5.15, 5.17 and 5.18) and proved that index-2does not exceed. The index bound is also true for Coulomb gauge (Theorem 5.8 and5.10). Furthermore we presented perturbation and solvability results for the coupledsystems and the perturbation index does not exceed two (Remark 5.13 and 5.21). Theelectromagnetic devices were inserted into the circuit as controlled current sources, butthe analysis showed that if using Lorenz gauge they, from the index point of view, didbehave like inductances. We concluded that in case of an index-1 configuration it isalways preferable to choose Lorenz gauge for the electromagnetic device.Next, we presented an approach for the calculation of an operating point. Based on thelinearity of the index-2 components (Lemma 5.12 and 5.20) the calculation of a consistentinitialization is possible by correcting an operating point by solving a linear system. Dueto the structure of the coupled system it is sufficient to start the numerical integrationwith the implicit Euler using an operating point to obtain a consistent initialization afterthe first time step.

134

6 Numerical Examples

In this chapter the different circuit models including memristors and electromagneticdevices are verified by some basic examples.The simulation software is written in Python and is an extension in the framework ofthe MECS (Multiphysical Electric Circuit Simulator) developed by the group of CarenTischendorf. The framework use for time integration a backward differentiation formulasimplementation with an adaptive order and step size control for index-2 differentialalgebraic equations with a properly stated leading term which is based on [Tis96].For the electromagnetic device simulation we integrate parts of the FIDES (Field DeviceSimulator) package of Sebastian Schops, see [Sch11], implemented for the magnetoqua-sistatic device simulation. FIDES is written in OCTAVE and integrated within theframework of the demonstrator platform of the CoMSON project (Coupled MultiscaleSimulation and Optimization in Nanoelectronics).The 3D Visualizations are obtained by Paraview.

6.1 Index Behavior of Field Problems

(a) 3D view.

0.5

mm

3 m

(b) Geometric dimensions.

Figure 6.1: Geometry of the copper bar.

Let us consider a copper bar used in [BCS12] with a cross-sectional area of 0.25 mm2

surrounded by air and discretized by the FIT, see Figure 6.1. The left contact is excitedby a sinusoidal source of the form v ptq “ sin p2πtq, the other contact is grounded.The simulations are carried out on the time interval r0 s, 0.5 ss by the implicit Eulerscheme with fixed step sizes h “ 8e-5 s, 4e-5 s, 2e-5 s, 1e-5 s.The numerical solution of the Lorenz (index-0) and Coulomb gauge (index-2) formula-tions of MGE are given in Figure 6.2(a) and 6.3(a). Both formulations provide solutionsas anticipated.

135

To analyze the sensitivity of the formulations we perturb the sinusoidal source v ptq bya small high-frequent noise

δ ptq “ 10´k sinp2 ¨ 10k`5πtq.We get the perturbed source vp ptq “ v ptq ` δ ptq. For the simulation, in this thesis, wehave chosen k “ 4. As expected, the numerical solution of the perturbed Lorenz-basedformulation is not affected, see Figure 6.2(b). On the other hand the solution of theindex-2 formulation suffers strongly from the perturbation, see Figure 6.3(b). The effect

0 1 2 3 4 5time [s] ×10−1

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

elec

tric

field

stre

ngth

[V]

h=1e-05h=2e-05h=4e-05h=8e-05

(a) unperturbed excited by v ptq.

0 1 2 3 4 5time [s] ×10−1

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

elec

tric

field

stre

ngth

[V]

h=1e-05h=2e-05h=4e-05h=8e-05

(b) perturbed excited by vp ptq.

Figure 6.2: Plot of a single component of the electric field using the Lorenz formulation (3.60).

occurs even for tiniest perturbations, that is, for very large k " 1. Moreover, the effectincreases with a reduction in step size, that is, it cannot be compensated by a finertemporal mesh. This is a typical index-2 behavior: The error increases while the step

0 1 2 3 4 5time [s] ×10−1

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

elec

tric

field

stre

ngth

[V]

h=1e-05h=2e-05h=4e-05h=8e-05

(a) unperturbed excited by v ptq.

0 1 2 3 4 5time [s] ×10−1

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

elec

tric

field

stre

ngth

[V]

h=1e-05h=2e-05h=4e-05h=8e-05

(b) perturbed excited by vp ptq.

Figure 6.3: Plot of a single component of the electric field using the Coulomb formulation (3.60).

size decreases. The index-2 problem is ill-conditioned but the perturbation error is not

136


propagated in time because the index-2 components enter only linearly. However, usinga step size control we should exclude the index-2 variables for the step size prediction,because the numerical error might be detected by the step size control and leads to anunprofitable reduction of the step size. The best case would be an unreasonably smallstep size whereas in the worst case the integration could completely fail.

6.2 Memristive Circuits

(a) Basic memristor circuit.

w

UD

d

(b) HP memristor circuit.

Figure 6.4: Memristor examples.

In this section we consider two models for the memristor to show that the MNA includingmemristor models (4.7) works properly, see Figure 6.4(a). The first example is the HPmemristor stated in [SSSW08], see Figure 6.4(b). The HP device is composed of a

0 1 2 3 4 5 6time [s] ×10−1

−1.5−1.0−0.5

0.00.51.01.5

curr

ent[

A]

×10−4

−1.0

−0.5

0.0

0.5

1.0

volta

ge[V

]currentvoltage

0 1 2 3 4 5 6time [s] ×10−1

0123456789

w/d

[1]

×10−1

(a) Voltage and current through and relativeboundary position of the device.

−1.0 −0.5 0.0 0.5 1.0voltage [V]

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

curr

ent[

A]

×10−4

(b) Lissajous curve: pinched hysteresis.

Figure 6.5: HP memristor with Roff “ 16e3 Ω and f “ 5 Hz.

thin titanium dioxide film between two electrodes containing a doped (D) region andan undoped (U) region and thus it behaves as a semiconductor. The application ofa voltage drop across the device moves the boundary between the two regions. Withelectric current passing in a given direction, the boundary between the two regions is

137

moving in the same direction. The total device length is d and the length of the doped

−0.8−0.6−0.4−0.2

0.00.20.40.60.8

curr

ent[

A]

×10−4

−1.0

−0.5

0.0

0.5

1.0

volta

ge[V

]currentvoltage

0 1 2 3 4 5 6time [s] ×10−2

1.0

1.1

1.2

1.3

1.4

1.5

w/d

[1]

×10−1


−1.0 −0.5 0.0 0.5 1.0voltage [V]

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

curr

ent[

A]

×10−4



region is denoted by w P r0, ds. The limits of the memristor resistance is given by Roff

and Ron for w “ 0 and w “ d. The dopant mobility is described by µV. The HP

−0.8−0.6−0.4−0.2

0.00.20.40.60.8

curr

ent[

A]

×10−4

−1.0

−0.5

0.0

0.5

1.0

volta

ge[V

]currentvoltage

0 1 2 3 4 5 6time [s] ×10−1

0123456789

w/d

[1]

×10−1


−1.0 −0.5 0.0 0.5 1.0voltage [V]

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

curr

ent[

A]

×10−4



memristor is modeled by the memristance

M pq, tq “ Roff

ˆ

1´ µVRon

d2q

˙

(6.1)

with Roff " Ron and the length of the doped region is given by

w “ µVRon

dq.

138


The simulations were carried out on the time interval r0 s, 0.6 ss by the implicit Eulerscheme using the parameters d “ 1e-8 m, Ron “ 1e2 Ω and µV “ 1e-13 mVs. ForFigure 6.5 and 6.6 we use as applied voltage source v ptq “ sin p2πftq and for Figure 6.7the applied voltage source is given by

v ptq “#

sin p2πftq2 , for t P r 0, 0.3s ,´ sin p2πftq2 , for t P p0.3, 0.6s.

Unfortunately, the results shown in Figure 2 of [SSSW08] do not fit the stated parameterstherein since the applied sinusoidal voltage source has in both cases a frequency of 5e-3Hz instead of 1e2 Hz. Nonetheless the results show the same qualitative behavior as ourresults here.

In fact the HP memristance (6.1) is a polynomial. Another memristance described by apolynomial is given in [BBBK10] and reads

M pq, tq “ r1 ` 3r3q2

with r1 “ 5 VA and r3 “ 1e4 VA3s2. The results in [BBBK10] are obtained by thecircuit given in Figure 6.4(a) in SPICE using a subcircuit to describe the behavior ofthe memristor.

0 1 2 3 4 5time [s]

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

curr

ent[

A]

×10−1

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

volta

ge[V

]

currentvoltage

(a) Voltage and current through the device.

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5voltage [V]

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

curr

ent[

A]

×10−1


Figure 6.8: Memristor with memristance described in [BBBK10].

Our simulations were carried out on the time interval r0 s, 5 ss by the implicit Eulerscheme and applied voltage source v ptq “ 13e-1 sin p2πtq. The results are given inFigure 6.8 and fit perfectly to the simulation results given in [BBBK10].

6.3 Coupled Field/Circuit Problems

We regard two interlocking open copper loops with a cross-sectional area of 1 mm2

surrounded by air and discretized by the FIT, see Figure 6.9(b). First we examine thebasic circuit given in Figure 6.9(a).

139

The simulations were carried out for Coulomb gauge on the time interval r0 s, f´1s bythe implicit Euler scheme using as supply voltage sources v ptq “ 1e-3 sin p2πftq withfrequency f .

(a) Basic circuit with two sources.

3.5 mm

0.5 mm

0.5

mm

1m

m

2m

m

0.5 mm

1 mm

(b) Geometric dimensions.

Figure 6.9: Two interlocking open copper loops.

For f “ 1 Hz we obtain the expected results since the static resistance of each opencopper loop is between 124e-6 Ω and 158e-6 Ω. The results are given in Figure 6.10. For

0.0 0.2 0.4 0.6 0.8 1.0time [s]

−8

−6

−4

−2

0

2

4

6

8

j 1[A

]

−8

−6

−4

−2

0

2

4

6

8

j 2[A

]

j1j2

(a) Light open copper loop.

0.0 0.2 0.4 0.6 0.8 1.0time [s]

−8

−6

−4

−2

0

2

4

6

8

j 3[A

]

−8

−6

−4

−2

0

2

4

6

8

j 4[A

]

j3j4

(b) Dark open copper loop.

Figure 6.10: Current through the open copper loops with f “ 1 Hz.

f “ 1e9 Hz we obtain the results given in Figure 6.11. The current through the darkopen copper loop is larger then the current through the light open copper loop. Thatbehavior, of course, results from the increasing frequency and arises from the proximityeffect, see [Ter43].The effect can be described as follows: When an alternating electric current flows throughan isolated conductor, it creates an associated alternating magnetic field around it, which

140


influences the distribution by electromagnetic induction of an electric current flowingwithin an electrical conductor.The alternating magnetic field induces eddy currents in adjacent conductors, alteringthe overall distribution of current flowing through them.

0.0 0.2 0.4 0.6 0.8 1.0time [s] ×10−9

−2

−1

0

1

2

j 1[A

]

×10−4

−2

−1

0

1

2

j 2[A

]

×10−4

j1j2

(a) Light open copper loop.

0.0 0.2 0.4 0.6 0.8 1.0time [s] ×10−9

−2

−1

0

1

2

j 3[A

]

×10−4

−2

−1

0

1

2

j 4[A

]

×10−4

j3j4

(b) Dark open copper loop.

Figure 6.11: Two interlocking open copper loops with f “ 1e9 Hz.

Eddy currents are electric currents induced in conductors when a changing magnetic fieldacts on the conductor and causes a circulating flow of current within the conductor, see[Ter43]. These currents are responsible for the skin effect in conductors. The skin effectis the tendency of an alternating electric current to distribute itself within a conductorwith the current density being largest near the surface of the conductor, decreasing atgreater depths, that is, the electric current flows mainly at the skin of the conductor.This effect is due to opposing eddy currents induced by the changing magnetic fieldresulting from the alternating current. Figure 6.12 demonstrates well the increasing

(a) Equal distribution for f “ 1 Hz. (b) Skin effect: surface currents forf “ 1e9 Hz.

Figure 6.12: Change in the distribution of the currents through the two interlocking open copper loops.

skin effect at increasing frequency for the basic circuit with two sources.

141

The second example circuit given in Figure 6.13 uses the two interlocking open copperloops of Figure 6.9(b), too. We choose R1 “ 9865e-6 Ω, R2 “ 140e-6 Ω, L “ 1e-2 H

R2

R2

R1 R1

L

v2 s

ptqv1 s

ptq(a) Equivalent circuit.

R1 R1

L

v2 s

ptqv1 s

ptqEM device

(b) Circuit with EM device.

Figure 6.13: More complex circuit with EM device.

and the supply voltage sources to be v ptq “ 1e-1 sin p2πftq with the frequency f . Thesimulations are carried out for Coulomb gauge on the time interval r0 s, 2f´1s by theimplicit Euler scheme. The results are given in Figure 6.14 and 6.15.

0.0 0.5 1.0 1.5 2.0time [s]

−1.0

−0.5

0.0

0.5

1.0

j 1[A

]

×101

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

j 2[A

]

j1j2

(a) f “ 1 Hz

0.0 0.5 1.0 1.5 2.0time [s] ×10−9

−1.0

−0.5

0.0

0.5

1.0

j 1[A

]

×101

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

j 2[A

]

×10−9

j1j2

(b) f “ 1e9 Hz

Figure 6.14: Equivalent circuit: Currents through the voltage sources.

In Figure 6.13(a) the EM device is replaced by two resistors with a resistance of R2.For f “ 1 Hz the results of both circuits are equal. This, of course, is not true forhigher frequencies. There the inductive behavior of the EM device plays a crucial role.Amongst others, inductive coupling occurs and affects the circuit strongly, see above.Furthermore due to the skin effect the effective resistance of the device is increased.

142


0.0 0.5 1.0 1.5 2.0time [s]

−1.0

−0.5

0.0

0.5

1.0

j 1[A

]

×101

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

j 2[A

]

j1j2

(a) f “ 1 Hz

0.0 0.5 1.0 1.5 2.0time [s] ×10−9

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

j 1[A

]

×10−2

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

j 2[A

]

×10−9

j1j2

(b) f “ 1e9 Hz

Figure 6.15: EM device circuit: Currents through the voltage sources.

Note that the equivalent circuit, Figure 6.13(a), is only for validating the low frequencyresults using the EM device.

6.4 Implementation Aspects

The simplest model in applied mathematics is asystem of linear equations. It is also by far themost important.

Gilbert Strang.

In every time integration step in transient simulation we have to solve linear systems.For effective solving of large linear systems iterative solvers and multigrid methods playan essential role since direct solvers are memory and time consuming and rapidly reachthe limit of applicability. For an introduction to iterative solvers and multigrid methodswe refer to [OST01, Saa03]. But most linear systems of the coupled simulation are not(directly) suitable for an iterative scheme since they are usually not symmetric, notpositive definite and not diagonally dominant.

However, we often find the following structure of the systems:

Az “„

A1 A2

A3 A4

ˆ

xy

˙

“ˆ

uv

˙

(6.2)

with A4 P Rn4ˆn4 , A1 P Rn1ˆn1 , n1 ! n4 and A4 being nonsingular and suitable foriterative solvers, while A1 is not. Applying the Schur complement S “ `

A1 ´ A2A´14 A3

˘

of the block A4 we achieve the two linear systems

A4y “ v ´ A3x (6.3)

143

and

Sx “ u´ A2A´14 v. (6.4)

The idea is to apply different linear solvers for solving the linear systems (6.3) and (6.4).To solve (6.4) we solve simultaneously n1 ` 1 linear systems of the dimension n4 inadvance, namely

A4w “ v (6.5)

and

A4W “ A3. (6.6)

Then (6.4) becomes

pA1 ´ A2Wq x “ u´ A2w (6.7)

For this, we solve (6.5) and (6.6) by an iterative solver. Then the linear system (6.7)can be solved by a direct solver to determine x P Rn1 .Finally, we solve the linear system (6.3) by an iterative solver, too. We obtain thesolution y P Rn4 and thus we have solved the original linear system (6.2).

For the field/circuit system using Lorenz gauge (5.18) the Jacobian of the BDF methodsfor the time integration can be decomposed into the blocks are given by:

A1 “»

–

α0

hACC

`

AJCe, t˘

AJC ` ARG`

AJRe, t˘

AJR AL AV

´AJLα0

hL pjL, tq 0

AJV 0 0

fi

fl

A2 “»

–

AE 0 0 00 0 0

0 0 0 0

fi

fl

A3 “

»

—

—

–

0 0 00 0 0

´ `

α0

hMuε `Mu

σ

˘

ΛuAJE 0 00 0 0

fi

ffi

ffi

fl

A4 “

»

—

—

–

I 0 ´ΛJu Kuν,dp"auq 0

0 α0

hrSuMu

εGurSuMu

ν 00

`

α0

hMuε `Mu

σ

˘

Gu Kuν,dp"auq α0

hMuε `Mu

σ

0 0 α0

hI ´I

fi

ffi

ffi

fl

Obviously, A1 coming from the MNA (4.6) is not (directly) suitable for an iterativesolver, whereas A4 coming from the MGE (3.65) using Lorenz gauge could be suitablefor iterative solvers, see Remark 3.41. This is motivated by the experience about the

144


use of iterative solvers for MGE especially for MQS and electroquasistatic devices, see[Hip98, CSvW96, Cle98, DW04, Cle05, Sch11].

In [BCK`11] we presented a systematic approach to reformulate the MNA (4.6) to beaccessible for iterative solvers. Here the main goal was to eliminate the zero entries onthe main diagonal of the Jacobian by manipulating the voltage sources. Nonetheless,this reformulation needs some effort and is usually not done within circuit simulationpackages.

6.5 Summary

We have shown that the extended models work as expected due to our theoretical findingsof the Chapters 3, 4 and 5. For the modified nodal analysis including memristors modelswe validated our model by using the HP memristor and an another memristance modelfrom literature. We obtained the same qualitative results for both modelsFor the electromagnetic device we showed the influence of the chosen gauge with respectto perturbations and we observed the predicted behavior.For the modified nodal analysis including electromagnetic device models we examinedtwo simple circuits. In both cases we investigated two interlocking open copper loopswith different frequencies for changing the applied potentials. For the low frequency casewe observed the expected resistance behavior of the device. For the high frequency casewe took note of the proximity and skin effect due to the inductive coupling.The combination of methods for solving the resulting linear systems were briefly out-lined.

145

Conclusion and Outlook

In this thesis, we presented the derivation of the modified nodal analysis including mem-ristor models and coupled electromagnetic device/circuit models and we investigated themodels in terms of the tractability index concept for differential-algebraic equations witha properly stated leading term.

We have derived new index results for circuits including memristors formulated asdifferential-algebraic equations with a properly stated leading term. We extended thewell-known topological index conditions of [Tis99, ET00] for the modified nodal analysisto circuits including memristors. The critical index-2 circuit configurations are loops ofonly capacitors and voltage sources and cutsets of only inductors and current sources.We concluded that, from index point of view, the memristors behave like resistors.

The electromagnetic devices were modeled by Maxwell’s equation in a potential for-mulation using the finite integration technique for the resulting spatial discretization.General properties of the discrete operators have been discussed. The spatial discretiza-tion leads to Maxwell’s grid equations, which were formulated in terms of potentials withincorporated boundary conditions using a new class of discrete gauge conditions in termsof the finite integration technique based on Lorenz gauge. The structural properties ofMaxwell’s grid equations were discussed and analyzed by the index concept to obtainknowledge about the stability of the solutions with respect to perturbations. The mainresult here is that the index depends on the chosen gauge condition. For Coulomb gaugewe obtain an index-2 differential-algebraic equation with a properly stated leading termwhereas for Lorenz gauge we achieve an ordinary differential equation.

The coupled electromagnetic device/circuit system was formulated as a differential-algebraic equation with a properly stated leading term. The coupling was realized bythe applied potential at the conductive contacts of the electromagnetic device and bythe current through it. In our case we have taken a different coupling approach than[DHW04, Ben06, BBS11, Sch11], where the coupling is realized using several conductormodels and applied as a source term. We generalized the well-known topological indexconditions for the modified nodal analysis to circuits refined by spatially resolved elec-tromagnetic devices modeled in case of using Lorenz gauge. In case of Lorenz gauge thecritical index-2 circuit configurations are loops of only capacitors and voltage sourcesand cutsets of only inductors, electromagnetic devices and current sources. For Coulombgauge we have shown that the index does not exceed two. The electromagnetic deviceswere inserted into the circuit as controlled current sources, but the analysis showed thatwhen applying Lorenz gauge they, from the index point of view, behave like inductances.

147

We concluded that in case of an index-1 circuit configuration it is always preferable tochoose Lorenz gauge for the electromagnetic device.

All considered differential-algebraic equations resulting from our fields of applicationshave a common structure such as a properly stated leading term, constant projectorsonto/along certain subspaces and linear index-2 variables. For that reason we investi-gated the differential-algebraic equations in a common differential-algebraic equationframework. For index-2 differential-algebraic equations we derived a local uniquelysolvability result and a perturbation estimation. To achieve these results we extendedthe well-known index reduction techniques for differential-algebraic equations without aproperly stated leading term to differential-algebraic equations with a properly statedleading term and exploited local uniquely solvability and perturbation results for index-1differential-algebraic equations with a properly stated leading term from literature. Inparticular we have proved that if the differential-algebraic equations with a properlystated leading term have index-2 then the differential-algebraic equation has perturba-tion index-2 as well. We extended the step-by-step approach by [Est00] for calculatingconsistent initial values for differential-algebraic equations with a properly stated lead-ing term using the linearity of the index-2 components. In addition we presented anapproach to calculate an operating point.

Some numerical examples were given to show that the models works as expected.

There are still a lot of unsolved problems and tasks to be tackled. For electromagneticdevices further sets of consistent boundary conditions for the coupled electromagneticdevice/circuit model and other models for nonlinear materials have to be considered.Beyond that it would be of great interest to combine existing models for electromag-netic devices, semiconductor devices and heating of devices. For this the next stepcould be to combine electromagnetic and semiconductor device models to study theinfluence of electromagnetic fields of the surrounding circuitry to semiconductor switch-ing elements. Then an index analysis of the coupled electromagnetic-semiconductordevice/circuit model would be an inevitable future step for the numerical simulation.Finally, heat effects could be studied by extending the models by heat conducting modelequations.

148

A Linear Algebra

We make use of several simple definitions and deductions from linear algebra in thisthesis and collect the results in this Chapter.

Lemma A.1. Let be A P Rnˆm. Then

ker A “ `

im AJ˘K

and im A “ `

ker AJ˘K

holds true.

Proof . We show both inclusions.

pĎq Let be x P ker A. Then

AJz P im AJ ñ 0 “ pAxqJ z “ xJAJz ñ x P `im AJ˘K

is true. For z P ker AJ we obtain

Ax P im A ñ 0 “ xJ`

AJz˘ “ pAxqJ z ñ z P `ker AJ

˘K.

pĚq Starting with

dim`

im AJ˘K ě dim ker A “ n´ dim im A ě n´ dim

`

ker AJ˘K

“ n´ `

m´ dim ker AJ˘ “ n´ dim im AJ “ dim

`

im AJ˘K

we get

dim ker A “ dim`

im AJ˘K

and dim im A “ dim`

ker AJ˘K.

Definition A.2. A (non-symmetric) matrix A P Rnˆn is called positive semidefinite ifand only if xJAx ě 0 and positive definite if and only xJAx ą 0 for all x ‰ 0, x P Rn.

From the definition above several simple results can be elementarily derived.

Lemma A.3. Let A P Rmˆm be positive definite and B P Rkˆm. Then

ker BABJ “ ker BJ and im BABJ “ im B

holds true.

149

Proof . At first we show ker BABJ “ ker BJ.

pĎq Let be x P ker BABJ. Then

xJBABJx “ 0 ñ yJAy “ 0, y “ BJx ñ BJx “ 0 ñ x P ker BJ,

since A is positive definite.

pĚq If x P ker BJ then BABJx “ 0 and hence x P ker BABJ.

Due to Lemma A.1 and we have

im BABJ “ `

ker BAJBJ˘K “ `

ker BJ˘K “ im B,

since AJ is positive definite.

A.1 Properties of Projectors

Fur euch ist es einfach. Ihr seid mit Projektorenaufgewachsen.

Roswitha Marz during the “DeutscheMathematiker-Vereinigung” conference,

Sept. 20, 2011, Cologne.

This section is devoted to basic definitions and results in projector calculus.

Definition A.4. The basics definitions for a projector are:

(i) A matrix Q P Rmˆm is called projector, if Q2 “ Q.

(ii) A projector Q P Rmˆm is called projector onto S Ď Rm, if im Q “ S.

(iii) A projector Q P Rmˆm is called projector along S Ď Rm, if ker Q “ S.

(iv) A projector Q P Rmˆm is called orthogonal projector if Q “ QJ.

Lemma A.5. Let Q P Rmˆm be a projector and P “ I´Q. Then:

(i) P is a projector.

(ii) x P im Q ô x “ Qx.

(iii) ker P “ im Q and ker Q “ im P.

(iv) ker Q‘ im Q “ Rm.

(v) If Q is nonsingular, then Q “ I.

150

A Linear Algebra

(vi) Let Q be a projector with P “ I´Q and im Q “ im Q. Then QQ “ Q and PP “ Phold true.

(vii) I` PEQ is nonsingular for all E P Rmˆm.

Proof . The proofs are straightforward.

(i) Q2 “ Q ô pI´ Pq2 “ I´ P ô P2 “ P.

(ii) x P im Q ô Dz P Rm : x “ Qz and Qx “ Qz ô x “ Qx.

(iii) x P ker P ñ Px “ 0 ñ x “ Qx ñ x P im Q. The other one is analog.

(iv) x “ Qx` Px, x P ker QX im Q ñ Qx “ 0, x “ Qx ñ x “ 0.

(v) Q2 “ Q ô Q´1Q2 “ I ô Q “ I.

(vi) Let be x P Rm. Then Qx P im Q “ im Q. Hence Qx “ QQx and Q “ QQ. We get

PP “ pI´Qq `I´Q˘ “ I´Q´Q´QQ “ I´Q “ P.

(vii) The inverse is given by I´ PEQ.

If Q is a projector then we call P “ I´Q the complementary projector.

Remark A.6. The product of two projectors is not necessarily a projector, too. Forthat we look at the projectors P1 and P2 given by

P1 “„

1 10 0

and P2 “„

1 01 0

.

Obviously P3 “ P1P2 is not a projector.

Lemma A.7. Let A P Rmˆn and Q P Rmˆm be a projector onto ker AJ. Then

ker QJ “ im A

holds true.

Proof . Using Lemma A.1 we obtain

ker QJ “ pim QqK “ `

ker AJ˘K “ im A.

Remark A.8. Instead of determining a projector Q along im A we can calculate QJ

onto ker AJ and betimes the computation of a projector onto a subspace is easier.

Lemma A.9. Let be A P Rnˆn. Then:

151

(i) Q P Rnˆn a projector onto im A implies QA “ A.

(ii) Q P Rnˆn a projector along ker A implies AQ “ A.

Proof . We get:

(i) im Q “ ker pI´Qq “ im A ñ pI´QqA “ 0 ñ QA “ A

(ii) ker Q “ im pI´Qq “ ker A ñ A pI´Qq “ 0 ñ AQ “ A

The next Lemma is motivated by [ET00].

Lemma A.10. Let A P Rmˆm be positive definite, B P Rkˆm, Q be a projector ontoker BJ and P “ I´Q. Then the matrix

H “ BABJ `QJQ

is positive definite and

HP “ BABJ “ PJH

holds true.

Proof . It is clear that H is positive semidefinite since it is the sum of two positivesemidefinite matrices. With that we get

zJHz “ 0 ô#

zJBABJz “ 0

zJQJQz “ 0

and we obtain z “ 0 by reason of z P ker BJ “ im Q and z P ker Q. Hence H is positivedefinite. The second statement follows immediately:

HP “ `

BABJ `QJQ˘

P “ BABJ “ PJ`

BABJ `QJQ˘ “ PJH

For computational aspects of projectors calculus we refer to [LMT13], where the projec-tors are determined by matrix decompositions.

A.2 Generalized Inverse

We report the (basic) definitions and relations of generalized inverses needed for ourconsiderations. A more detailed look on this topic is provided by, for example, [BIG03,BO71].

152

A Linear Algebra

Definition A.11. With A´ P Rnˆm we denote a pseudoinverse and t1, 2u-inverse ofA P Rmˆn if

AAÁ “ A and AÁA´ “ A´

are fulfilled.

Lemma A.12. A pseudoinverse A´ P Rnˆm exists for every A P Rmˆn.

Proof . For every A P Rmˆn it exist nonsingular matrices S P Rmˆm and T P Rnˆn with

SAT “„

I 00 0

ô A “ S´1

„

I 00 0

T´1

with I P Rrˆr and r “ rank A. The matrix

A´ “ T

„

I XY YX

S

fulfills all necessary properties, where Y P Rpm´rqˆr, X P Rrˆpn´rq are arbitrarily, see[BO71].

Lemma A.13. The matrices AA´ P Rmˆm and AÁ P Rnˆn are projectors onto im Aand along ker A.

Proof . The projector properties are clear due to the definition of the pseudoinverse. Itremains to show im AA´ “ im A and ker AÁ “ ker A. We show the first identity. Wehave:

pĎq x P im AA´ ñ x “ AA´z ñ x “ Ay ñ x P im A

pĚq x P im A ñ x “ Az ñ x “ AAÁz ñ x “ AAý ñ x P im AA´

The second identity is completely analog.

Let R P Rnˆn be a projector onto im A and P P Rmˆm a projector along ker A.

Theorem A.14. Let R P Rnˆn be a projector onto im A and P P Rmˆm a projectoralong ker A. The choice

(i) AÁ “ P

(ii) AA´ “ R

is always possible.

153

Proof . Let be I P Rrˆr and r “ rank A. It exists nonsingular matrices S P Rmˆm andT P Rnˆn with

A´ “ T

„

I XY YX

S, A´A “ T

„

I 0Y 0

T´1, and AA´ “ S´1

„

I X0 0

S,

see proof of Lemma A.12.(i) Let be z P ker P “ ker A´A. Then y P ker P1 “ ker P2 with

P1 “ T´1PT, P2 “„

I 0Y 0

and y “ T´1z.

The matrices P1 and P2 are projectors with the same nullspace. We have

P2y “ 0 ñ p0, y2q P ker P2 with y “ py1, y2qand hence we obtain

P1 “„

P1,1 0P1,2 0

and P21 “

„

P21,1 0

P1,2P1,1 0

.

Since P1 is a projector we conclude that P1,1 is a projector, too, and P1,2 “ P1,2P1,1.Assume P1,1 ‰ I, that is, P1,1 is singular. Then there exist y1 ‰ 0 with P1,1y1 “ 0 andP1y “ 0 with y “ py1, y2q. With y1 ‰ 0 we obtain

P2y “ˆ

y1

Yy1

˙

‰ˆ

00

˙

which is a contradiction to the property that both nullspaces equals. Hence P1,1 “ I and

T´1PT “„

I 0P1,2 0

with Y “ P1,2.

(ii) Let be z P im R “ im AA´. Then y P im R1 “ im R2 with

R1 “ SRS´1, R2 “„

I X0 0

and y “ Sz.

The matrices R1 and R2 are projectors with the same image. We have

y P im R2 ñ y “ py1, 0q with y “ py1, y2qand hence we obtain

R1 “„

R1,1 R1,2

0 0

and R21 “

„

R21,1 R1,1R1,2

0 0

.

154

A Linear Algebra

Since R1 is a projector we conclude that R1,1 is a projector, too, and R1,2 “ R1,1R1,2.Assume R1,1 ‰ I, that is, R1,1 is singular. Then there exist y1 ‰ 0, y1 P Rr, withR1,1y1 “ 0. Thus y1 R im R1,1 and hence py1, 0q R im R1. But py1, 0q P im R2 since

ˆ

y1

0

˙

“„

I X0 0

ˆ

y1

0

˙

.

That is a contradiction to the property that both images equals. Hence R1,1 “ I and

SRS´1 “„

I R1,2

0 0

with X “ R1,2.

A proof of Theorem A.14 is already given in [BIG03], Chapter 2, Theorem 12. But theproof given above provides a way to construct the pseudoinverse A´ explicitly with thespecial choice above.

Lemma A.15. Let R P Rnˆn be a projector onto im A and P P Rmˆm a projector alongker A. The pseudoinverse A´ together with

AÁ “ P and AA´ “ R

is uniquely determine.

Proof . Let B be a pseudoinverse of A too. Then:

B “ BAB “ BR “ BAA´ “ PA´ “ AÁA´ “ A´

Definition A.16. With A` P Rnˆm we denote the Moore-Penrose pseudoinverse andt1, 2, 3, 4u-inverse of A P Rmˆn if

AAÀ “ A AÀA` “ A`

AÀ “ `

AÀ˘J

AA` “ `

AA`˘J

are fulfilled.

The Moore-Penrose pseudoinverse is a special pseudoinverse. In contrast to a pseudoin-verse we require also that AÁ and AA´ are orthogonal projectors

Lemma A.17. A Moore-Penrose pseudoinverse A` exist for every A P Rmˆn.

Proof . Let A “ UΣVJ be a singular value decomposition. The Moore-Penrose pseu-doinverse is given by

A` “ V

„

D´1 00 0

UJ and Σ “„

D 00 0

.

The necessary properties follow immediately.

155

An essential difference between the pseudoinverse A´ and the Moore-Penrose pseudoin-verse A` is the uniqueness of the latter.

Lemma A.18. The Moore-Penrose pseudoinverse A` of A P Rmˆn is uniquely deter-mine.

Proof . Let B be also Moore-Penrose pseudoinverse of A. Then

B “ BAB “ B pABqJ “ BBJAJ “ BBJ`

AAÀ˘J “ BBJAJ

`

AÀ˘J

“ B pABqJ ÀÀ˘J “ BABAA` “ BAA` “ pBAqJ AÀA`

“ pBAqJ ÀÀ˘J

A` “ AJBJAJ`

A`˘J

A` “ pABAqJ À`˘J A`

“ AJ`

A`˘J

A` “ `

AÀ˘J

A` “ AÀA` “ A`

is valid.

Lemma A.19. The matrices AA` P Rmˆm and AÀ P Rnˆn are projectors along ker AJ

and onto im AJ.

Proof . The projector properties is clear due to the definition of the pseudoinverse. Itremains to show ker AA` “ ker AJ and im AÀ “ im AJ. We show the first identity.We have

ker AA` “ ker`

AA`˘J “ `

im AA`˘K “ pim AqK “ ker AJ.

The second identity is completely analog.

Remark A.20. If A P Rnˆn is nonsingular, the pseudoinverse and the inverse coincide.

Lemma A.21. Let A P Rmˆm be positive definite and B P Rkˆm. Then

im B`B “ im B`BABJ

holds true.

Proof . We show both inclusions.

(Ă) Let be x P im B`B. Then there exists y P Rm such that x “ B`By and z P im Bwith z “ By. Lemma A.3 leads to z P im BABJ and there exists u P Rk such thatz “ BABJu. Hence we obtain x “ B`BABJu and x P im B`BABJ.

(Ą) Let be x P im B`BABJ. Then there exists y P Rk such that x “ B`BABJy.Choosing z “ ABJy we obtain x “ B`Bz and x P im B`B.

For computational aspects of generalized inverses calculus we refer to [LMT13], wherethe generalized inverses a determine by matrix decompositions.

156

B Graph Theory

In this section we want to introduce elementary basics derived from the theory of graphsand digraphs. For more details we refer the reader to [Die05]. First, we start with somebasic notation and definitions. Roughly speaking, a graph is a set of edges and the endsof the edges are called nodes. If all edges own an orientation then the graph is called adigraph. Let N be a set. Then |N | P N is the number of elements in N .

Definition B.1 (graph, node, edges). A graph G is a tuple of finite sets G :“ pN , Eqsuch that E Ď N ˆN with |N | , |E | ă 8. We call an element of the set N node and ofthe set E edge. In general each edge corresponds to an unsorted tuple of nodes denotedby e “ pn1, n2q and e “ pn2, n1q, respectively, with e P E and n1, n2 P N . We call n anode of G if n P N and e an edge of G if e P E .

A graph, where edges correspond to an unsorted tuple of nodes, is called undirectedgraph. We say that two nodes n1 and n2 of G are adjacent if either e “ pn1, n2q ore “ pn2, n1q are edges of G. In case of e “ pn1, n2q we say that n1 is the front node andn2 is the back node of edge e. A node n of G is called incident to an edge e of G if n isthe front or back node of e. Two edges e1 and e2 of G are called incident if these edgeshave one common node n of G and an edge e of G is called incident with a node n of Gif the node n is the front or back node of the edge e .

A common approach to illustrate a graph is drawing a dot for each node and joining twoof dots by a line if there exists an edge between these two dots. How to draw the dotsand edges is considered irrelevant because all relevant information is the node-to-edge-relation. Hence the representation of a graph is not unique.

For further investigations we exclude the possibility to have an edge with the same frontand back node.

Definition B.2 (digraph). A digraph G is a graph, where each edge corresponds to asorted tuple of nodes. We say that the edges are orientated.

In case of digraphs the representation of edges are considered as arrows instead of lines.Each digraph G we can assign an undirected graph by dropping the edge orientation. Ifwe speak about graphs we include digraphs by the assigned undirected graph.

Definition B.3 (path). A set of n edges te1, . . . , enu Ď E of a graph G is called a pathbetween n1 and n2 if:

(i) the edges ei and ei`1 are incident, i P t1, . . . , n´ 1u

157

(ii) each node is incident to at most two edges

(iii) the nodes n1, n2 belong to exactly one edge of the set

Definition B.4 (Connected graph). A graph is called a connected graph if there existsat least one path between any two nodes of the graph. Otherwise we call the graphdisconnected.

For defining loops, trees and cutsets of a graph we need the following definition of asubgraph.

Definition B.5 (subgraph). A graph G 1 :“ pN 1, E 1q is called a subgraph of G if N 1 Ď N ,E 1 Ď E and E 1 Ď N 1 ˆN 1.The difference graph GzG 1 is given by GzG 1 “ pN , EzE 1q andfor e P E we define pGzG 1q Y teu “ pN , pEzE 1q Y teuq.Next we can define loops, trees and cutsets of a graph.

Definition B.6 (loop). A subgraph G 1 of a connected graph G is called a loop if it isconnected and precisely two edges of it are incident with each node.

Definition B.7 (tree). A subgraph G 1 of a connected graph G is called a tree if:

(i) G 1 is connected

(ii) G 1 contains all nodes of G(iii) G 1 has no loops

It should be mentioned that we can construct a tree for each connected graph. Further-more each tree of a connected graph with |N | nodes consists of exactly |N | ´ 1 edges,see [Die05] Proposition 1.5.3 and 1.5.6.

Definition B.8 (cutset). A subgraph G 1 of a connected graph G is called a cutset if:

(i) GzG 1 is disconnected

(ii) For every e1 P E 1 the graph pGzG 1q Y te1u is connected

Now we will focus on digraphs and combine some linear algebra with graph theory by in-troducing the so-called incidence matrix for digraphs. We obtain a matrix representationfor a graph, which shows the relationship between nodes and edges.

Definition B.9 (incidence matrix). Let a digraph G with |N | nodes and |E | edges be

given. The incidence matrix denoted by Aa P t´1, 0, 1u|N |ˆ|E| is defined as Aa “ paijqwith

aij “

$

’

&

’

%

1 if the edge j leaves the node i,

´1 if the edge j enters the node i,

0 else.

158

B Graph Theory

By definition of the incidence matrix Aa of a connected digraph it is easy to see thatthe rows are linear dependent. To be more specific the sum of all rows of Aa equalszero. This is caused by the fact that each column contains exactly one 1 and one ´1and all other entries are zero. This becomes obvious if one observes that each columncorresponds to exactly one edge and each edge has two incident nodes. Hence one rowof the incidence matrix can be neglected in order to describe the graph. That nodecorresponding to the neglected row is called reference node and can be chosen freely.Erasing one row of Aa we obtain the so-called reduced incidence matrix A. In literaturethe reduced incidence matrix A is often called only incidence matrix A. In our case wewill name the matrix A incidence matrix too.

e2 e3 e4

n3 n4

n2n1

e1

e5

(a) undirected graph

e2 e3 e4

n3 n4

n2n1

e1

e5

(b) digraph

Figure B.1: An undirected graph and digraph with four nodes and five edges.

Example B.10. Regarding the graph G “ pN , Eq in Figure B.1(a) the set of nodes isgiven by N “ tn1, n2, n3, n4u and the set of edges by E “ te1, e2, e3, e4, e5u. The graphG is the undirected version of the digraph in Figure B.1(b). Obviously the graph G isconnected. A loop is given by the edges te1, e4, e3u, but not by the edges te2, e5u. Thelast set of edges builds a path starting in node n1 and ending in node n4. In this casethe set describes a cutset, too. A tree is given by the edges te1, e3, e5u. Choosing thenode n4 as reference node the incidence matrices are

Aa “

»

—

—

–

1 ´1 1 0 0´1 0 0 ´1 00 1 0 0 ´10 0 ´1 1 1

fi

ffi

ffi

fl

and A “»

–

1 ´1 1 0 0´1 0 0 ´1 00 1 0 0 ´1

fi

fl

respectively.

In the following we collect some facts about incidence matrices.

Theorem B.11. The incidence matrix A of a connected graph G with |N | nodes has|N | ´ 1 linear independent rows.

159

Theorem B.12. A subgraph G 1 of a connected graph G with |E 1| edges has loops if andonly if the columns of the incidence matrix A corresponding to these |E 1| edges are lineardependent.

Theorem B.13. Let A be the incidence matrix of a connected graph G with |N | nodes.Then |N | ´ 1 columns of A are linear independent if and only if the edges of thesecolumns form a tree.

We refer the reader to Appendix A.1 in [Tis04] for the proofs.

Remark B.14 (Incidence matrices). We note the following:

(i) An incidence matrix AX of a subgraph has full column rank if and only if there isno loop (in the subgraph), that is, no X-loop, see Theorem B.12.

(ii) Let“

AX AY

‰

denote the incidence matrix of a connected graph. Then AJX hasfull column rank if and only if there is a spanning tree of elements from AX, thatis, no Y-cutset, see Theorem B.13.

160

C Auxiliary Calculations

C.1 Topological Projectors

Let the network edges be sorted in such a way that the (reduced) incidence matrix Ahas the form

A “ “

AC AR AL AV AI

‰

,

where the index stands for capacitive, (extended) resistive, (extended) inductive, (ex-tended) voltage source and current source edges, respectively.In order to describe different circuit configuration in more detail we will introduce someuseful projectors.We denote by

QC, QC´V, QVĆ, QRĆV and QCRV

projectors onto

ker AJC, ker QJCAV, ker AJ

VQC, ker AJ

RQCQVĆ and ker

“

AC AR AV

‰J,

respectively. All complementary projectors will be denoted by P “ I ´ Q with corre-sponding subindex.In the following we show that QCRV “ QCQVĆQRĆV is a valid construction. Thatspecial construction goes back to [ET00], here it is slightly extended them to an enlargeclass of network edges.At first we introduce some notation and results concerning special cutsets and loops tomotivate the projectors above.

Definition C.1 (LI-cutset). A cutset is called LI-cutset if and only if the cutset contains(extended) inductors and current sources only.

Lemma C.2 (LI-cutsets). Let a connected circuit be given. The circuit does not containan LI-cutset if and only if

(i) the matrix“

AC AR AV

‰

has full row rank or

(ii) the projector QCRV is equal to the zero matrix.

Proof . See Lemma 1.2 in [Tis04].

161

Definition C.3 (CV-loop). A loop is called CV-loop if and only if the loop containscapacitors and (extended) voltage sources only.

Lemma C.4 (CV-loops). Let QC be a projector onto ker AJC. The circuit does notcontain a CV-loop with at least one (extended) voltage source if and only if

(i) the matrix QJCAV has full column rank or

(ii) the projector QC´V is equal to the zero matrix.

Proof . See Lemma 1.3 in [Tis04].

Next we construct the projector QCRV. For that we need the following lemmata.

Lemma C.5. The relations

(i) im PC Ă ker PVĆ

(ii) im PVĆ Ă ker PRĆV

(iii) im PC Ă ker PRĆV

hold true.

Proof . Straightforward computations show the results.

(i) x P ker QC ñ AJV

QCx “ 0 ñ x P im QVĆ

(ii) x P ker QVĆ ñ AJR

QCQVĆx “ 0 ñ x P im QRĆV

(iii) x P ker QC

(i)ñ AJR

QCQVĆx “ 0 ñ x P im QRĆV

Corollary C.6. The relations

(i) PVĆPC “ 0 ô QVĆQC “ QC `QVĆ ´ I

(ii) PRĆVPC “ 0 ô QRĆVQC “ QC `QRĆV ´ I

(iii) PRĆVPVĆ “ 0 ô QRĆVQVĆ “ QVĆ `QRĆV ´ I

hold true.

Lemma C.7. QCQVĆ is a projector.

Proof . Using Corollary C.6 we get

`

QCQVĆ

˘2 “ QC

`

QC `QVĆ ´ I˘

QVĆ “ QCQVĆ.

162


Lemma C.8. QCQVĆQRĆV is a projector.

Proof . Using Corollary C.6 and Lemma C.7 we get

`

QCQVĆQRĆV

˘2 “ QCQVĆ

`

QC `QRĆV ´ I˘

QVĆQRĆV

“ QCQVĆQRĆVQVĆQRĆV

“ QCQVĆ

`

QVĆ `QRĆV ´ I˘

QRĆV

“ QCQVĆQRĆV.

Lemma C.9. The relations im QCQVĆQRĆV Ă im QC hold true.

Lemma C.10. The matrix QCQVĆQRĆV is a projector onto ker“

AC AR AV

‰J.

Proof . We have to show two inclusions.

pĎq Let be x P im QCQVĆQRĆV. Then x P im QC “ ker AJC. We obtain x P ker AJR

since im QRĆV “ ker AJR

QCQVĆ. Due to im QVĆ “ ker AJV

QC we get x P ker AJV

and we conclude im QCQVĆQRĆV Ă ker“

AC AR AV

‰J.

pĚq Let be x P ker“

AC AR AV

‰J. Then we get x P im QC and also x P ker AJ

VQC.

Consequently, we gain x P im QVĆ and thus x P im QCQVĆ. From x P ker AJR

we obtain x P ker AJRQCQVĆ and hence x P im QRĆV. Accordingly, we achieve

x P im QCQVĆQRĆV and therefore ker“

AC AR AV

‰J Ă im QCQVĆQRĆV.

Lemma C.11. The relation ker QC Ă ker QCRV hold true.

Proof . We use Lemma C.5. Let be x P ker QC. Then x P im QVĆ and we achievex P ker QCQVĆ. Therefore x P im QRĆV and thus x P ker QCRV.

Corollary C.12. The matrix

QCRV “ QCQVĆQRĆV

is a projector onto ker“

AC AR AV

‰Jand the relation QCRVQC “ QCRV holds true.

C.2 Electric Network

Lemma C.13. Let Assumption 4.4 and 4.9 be fulfilled. For the DAE (4.8) we get

W0 py, tq “

»

—

—

–

QJC ´QJ

CAMM pqM, tq´1 0 00 0 0 00 0 0 00 0 0 I

fi

ffi

ffi

fl

.

163

Proof . We compute a projector along im G0 py, tq. In order to determine such a pro-jector we calculate a projector onto ker G0 py, tqJ, see Remark A.8, with

G0 py, tqJ “

»

—

—

—

–

ACC`

AJC, t˘J

AJC 0 0 0

AJM M pqM, tqJ 0 0

0 0 L pjL, tqJ 00 0 0 0

fi

ffi

ffi

ffi

fl

,

see (4.9). Let be z P ker G0 py, tqJ. This it true if and only if

ze P im QC

´M pqM, tq´J AJMze,t “ zqM

zjL “ 0

hold true, using Assumption 4.4, 4.9 and Lemma A.3. We can choose a projector ontoker G0 py, tqJ and along im G0 py, tq by

W0 py, tqJ “

»

—

—

–

QC 0 0 0

´M pqM, tq´J AJMQC 0 0 00 0 0 00 0 0 I

fi

ffi

ffi

fl

and

W0 py, tq “

»

—

—

–

QJC ´QJ

CAMM pqM, tq´1 0 00 0 0 00 0 0 00 0 0 I

fi

ffi

ffi

fl

,

respectively.


W1 “

»

—

—

–

QJCRMV 0 0 00 0 0 00 0 0 00 0 0 QJ

C´V

fi

ffi

ffi

fl

.

Proof . We compute a projector along im G1 py, tq. On this we determine a projectoronto ker G1 py, tqJ, see Remark A.8. Hereby we investigate

G1 py, tqJ “

»

—

—

–

G pe, tqJ ´QJCAM ´QJ

CAL QJCAV

AJM M pqM, tqJ 0 0

0 0 L pjL, tqJ 0AJV 0 0 0

fi

ffi

ffi

fl

,

164



AJCe, t˘J

AJC ` ARG`

AJRe, t˘J

AJRQC, see (4.13).

Let be z P ker G1 py, tqJ. That is true if and only if

´

ACC`

AJCe, t˘J

AJC `QJCARG

`

AJRe, t˘J

AJR

¯

ze ´QJCAMzqM

`QJCAVzjV “ 0 (C.1)

´M pqM, tq´J AJMze “ zqM(C.2)

zjL “ 0

AJVze “ 0 (C.3)

hold true, taking Assumption 4.4 and 4.9 into account. Left-multiplication of (C.1) byQJ

C yields

QJCARG

`

AJRe, t˘J

AJRze ´QJCAMzqM

`QJCAVzjV “ 0

and subtraction of (C.1) leads to ze P im QC by applying Lemma A.3. Using ze P im QC

and (C.2) we can rewrite (C.1) as

QJCARG

`

AJRe, t˘J

AJRQCze `QJCAMM pqM, tq´J AJMQCze `QJ

CAVzjV “ 0.

Left-multiplying that by zJe leads to ze P ker“

AR AM

‰Jby taking (C.3) into account.

Together with (C.3) we attain ze P im QCRMV. From (C.2) we obtain zqM“ 0 and (C.1)

yields

QJCAVzjV “ 0 and zjV P im QC´V,

that is, we can choose a projector onto ker G1 py, tqJ and along im G1 py, tq by

WJ1 “

»

—

—

–

QCRMV 0 0 00 0 0 00 0 0 00 0 0 QC´V

fi

ffi

ffi

fl

and W1 “

»

—

—

–

QJCRMV 0 0 00 0 0 00 0 0 00 0 0 QJ

C´V

fi

ffi

ffi

fl

,

respectively.


N1 py, tq “ tz P Rn|QCze P im QCRMV, zjV P im QC´V, zqM“ 0,

PCze “ ´HC

`

AJCe, t˘´1


)

.

Proof . Let be z P ker G1 py, tq. This is holds if and only if

ACC`

AJCe, t˘

AJC ` ARG`

AJRe, t˘

AJRQCze ` AMzqM` AVzjV “ 0, (C.4)

M pqM, tq´1 AJMQCze “ zqM(C.5)

165

L pjL, tq´1 AJL QCze “ zjL

AJVQCze “ 0, (C.6)

are valid, see (4.13) and using Assumption 4.4 and 4.9. Left-multiplication of (C.4) byzJe QJ

C and inserting (C.5) give rise to

zJe QJCARG

`

AJRe, t˘

AJRQCze ` zJe QJCAMM pqM, tq´1 AJMQCze ` zJe QJ

CAVzjV “ 0

Utilizing (C.6) leads to QCze P ker“

AR AM

‰Jby using Lemma A.3. We attain zqM

“ 0and together with (C.6) to QCze P im QCRMV. Left-multiplication of (C.4) by QJ

C resultsin

QJCAVzjV “ 0 and zjV P im QC´V.

Moreover we can reduce (C.4) to

ACC`

AJCe, t˘

AJCze ` AVzjV “ 0

which can be rewritten as

HC

`

AJCe, t˘

PCze ` AVzjV “ 0,

where HC

`

AJCe, t˘ “ ACC

`

AJCe, t˘

AJC ` QJCQC is positive definite, see Lemma A.10.

Hence we deduce that

QCze P im QCRMV

zjV P im QC´V

zqM“ 0

PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV


hold true and yields the representation

N1 py, tq “ tz P Rn|QCze P im QCRMV, zjV P im QC´V, zqM“ 0,

PCze “ ´HC

`

AJCe, t˘´1


)

.

C.3 Field/Circuit System using Coulomb Gauge

Lemma C.16. Assume Assumption 4.4, 3.31 and Property 3.32 to be fulfilled. For theDAE (5.2) we get

W0 “

»

—

—

—

—

—

—

—

—

–

QJC 0 0 0 0 0 0

0 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

166


Proof . We compute a projector along im G0 py, tq. In order to determine such a pro-jector we calculate a projector onto ker G0 py, tqJ, see Remark A.8, with

G0 py, tqJ “

»

—

—

—

—

—

—

—

—

—

–

ACC`

AJCe, t˘J

AJC 0 0 0 0 ´AEΛJu Muε 0

0 L pjL, tqJ 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 ´rSuMuε 0

0 0 0 0 0 0 I0 0 0 0 0 Mu

ε 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

see (5.3), using Assumption 3.31 and Property 3.32. Let be z P ker G0 py, tqJ. This isvalid if and only if

ze P im QC

zjL “ 0

z"au“ 0

z"πu“ 0

hold true, using Assumption 4.4 and Lemma A.3. We can choose a projector ontoker G0 py, tqJ and along im G0 py, tq by

WJ0 “

»

—

—

—

—

—

—

—

—

–

QC 0 0 0 0 0 00 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and W0 “

»

—

—

—

—

—

—

—

—

–

QJC 0 0 0 0 0 0

0 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

respectively.

Lemma C.17. Let Assumption 4.4, 3.31 and Property 3.32 be true. For the DAE (5.2)we get

W1 “

»

—

—

—

—

—

—

—

—

–

QJCRV 0 0 ´QJ

CRVAE 0 0 00 0 0 0 0 0 00 0 QJ

C´V 0 0 0 00 0 0 0 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

167

Proof . We compute a projector along im G1 py, tq. On this we determine a projectoronto ker G1 py, tqJ, see Remark A.8. Hereby we investigate

G1 py, tqJ “

»

—

—

—

—

—

—

—

—

–

G pe, tqJ ´QJCAL QJ

CAV 0 0 ÁEΛJu Muε ´QJ

CAEΛJu0 L pjL, tqJ 0 0 0 0 0

AJV 0 0 0 0 0 0AJE 0 0 I 0 0 0

0 0 0 0 0 ´rSuMuε ´rSu

0 0 0 0 0 0 I0 0 0 0 0 Mu

ε 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

where G pe, tq “ ACC`

AJCe, t˘

AJC`QJCARG

`

AJRe, t˘

AJR, see (5.7), using Assumption 3.31

and Property 3.32. Let be z P ker G1 py, tqJ. This is valid if and only if

´

ACC`

AJCe, t˘J

AJC `QJCARG

`

AJRe, t˘J

AJR

¯

ze `QJCAVzjV “ 0 (C.7)

zjL “ 0

AJVze “ 0 (C.8)

zjE “ ÁJEze

z"au“ 0

z"πu“ 0

hold true, taking Assumption 4.4 into account. Left-multiplication of (C.7) by QJC yields

QJCARG

`

AJRe, t˘J

AJRze `QJCAVzjV “ 0 (C.9)

and subtraction from (C.7) leads to

ACC`

AJCe, t˘J

AJCze “ 0.

Hence we obtain ze P im QC due to Lemma A.3. Left-multiply (C.9) by zJe and taking(C.8) into account leads to ze P im QCRV. From (C.9) follows that zjV P im QC´V. Hencewe deduce

ze P im QCRV

zjL “ 0

zjV P im QC´V

zjE “ ÁJEQCRVze

z"au“ 0

z"πu“ 0

168


and we can choose a projector onto ker G1 py, tqJ and along im G1 py, tq by

WJ1 “

»

—

—

—

—

—

—

—

—

–

QCRV 0 0 0 0 0 00 0 0 0 0 0 00 0 QC´V 0 0 0 0

´AJEQCRV 0 0 0 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and

W1 “

»

—

—

—

—

—

—

—

—

–

QJCRV 0 0 ´QJ

CRVAE 0 0 00 0 0 0 0 0 00 0 QJ

C´V 0 0 0 00 0 0 0 0 0 00 0 0 0 I 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

respectively.


N1 py, tq “



`

AJCe, t˘´1



(

.

Proof . Let be z P ker G1 py, tq. This is valid if and only if

`

ACC`

AJCe, t˘

AJC ` ARG`

AJRe, t˘

AJRQC

˘

ze ` AVzjV “ 0 (C.10)


AJVQCze “ 0 (C.11)

zjE “ 0

ΛuAJEze ´Guzφu “ z"πu

ΛuAJEQCze ´Guzφu “ z"au

hold true, see (5.3), using Assumption 4.4, 3.31 and Property 3.32. Left-multiplicationof (C.10) by zJe QJ

C and taking advantage of (C.11) results in QCze P ker AJR due toLemma A.3. From (C.11) we attain QCze P im QCRV. With this (C.10) reduces to

ACC`

AJCe, t˘

AJCze ` AVzjV “ 0 (C.12)

169


HC

`

AJCe, t˘

PCze ` AVzjV “ 0,

where HC

`

AJCe, t˘ “ ACC

`

AJCe, t˘


Left-multiplication of (C.12) by QJC leads to zjV P im QC´V. Hence we deduce that

QCze P im QCRV

zjV P im QC´V

zjE “ 0

PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV


ΛuAJEze ´Guzφu “ z"πu

ΛuAJEQCze ´Guzφu “ z"au

hold true and we obtain

N1 py, tq “



`

AJCe, t˘´1



(

.

C.4 Field/Circuit System using Lorenz Gauge


W0 “

»

—

—

—

—

—

—

—

—

–

QJC 0 0 0 0 0 0

0 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Proof . We compute a projector onto ker G0 py, tq. On this we determine a projectoralong ker G0 py, tqJ, see Remark A.8. Hereby we investigate

G0 py, tqJ “

»

—

—

—

—

—

—

—

—

—

–

ACC`

AJCe, t˘J

AJC 0 0 0 0 ´AEΛJu Muε 0

0 L pjL, tqJ 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 rSuMuεGu ´rSuMu

ε 00 0 0 0 0 0 I0 0 0 0 0 Mu

ε 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

170


see (5.19), using Assumption 3.31 and Property 3.32. Let be z P ker G0 py, tqJ. This isvalid if and only if

ze P im QC

zjL “ 0

z"au“ 0

z"πu“ 0

zφu “ 0

hold true, due to Lemma A.3 and Assumption 4.4. Hence we can choose a projectoronto ker G0 py, tqJ and along im G0 py, tq by

WJ0 “

»

—

—

—

—

—

—

—

—

–

QC 0 0 0 0 0 00 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and W0 “

»

—

—

—

—

—

—

—

—

–

QJC 0 0 0 0 0 0

0 0 0 0 0 0 00 0 I 0 0 0 00 0 0 I 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Lemma C.20. Assume Assumption 4.4, 3.31 and Property 3.32 to be fulfilled. For theDAE (5.18) we get

W1 “

»

—

—

—

—

—

—

—

—

–

QJCRV 0 0 ´QJ

CRVAE 0 0 00 0 0 0 0 0 00 0 QJ

C´V 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Proof . We compute a projector along im G1 py, tq. On this we determine a projectorW1 py, tqJ along ker G1 py, tqJ, see Remark A.8, where

G1 py, tqJ “

»

—

—

—

—

—

—

—

—

–

G pe, tqJ ´QJCAL QJ

CAV 0 0 ´AEΛJu Muε ´QJ

CAEΛJu0 L pjL, tqJ 0 0 0 0 0

AJV 0 0 0 0 0 0AJE 0 0 I 0 0 0

0 0 0 0 rSuMuεGu ´rSuMu

ε 00 0 0 0 0 0 I0 0 0 0 0 Mu

ε 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

171

see (5.20), with G pe, tq “ ACC`

AJCe, t˘

AJCÀRG`

AJRe, t˘

AJRQC, using Assumption 3.31

and Property 3.32. Let be z P ker G1 py, tqJ. This is valid if and only if

´

ACC`

AJCe, t˘J

AJC `QJCARG

`

AJRe, t˘J

AJR

¯

ze `QJCAVzjV “ 0 (C.13)

zjL “ 0

AJVze “ 0 (C.14)

zjE “ ÁJEze

z"au“ 0

z"πu“ 0

zφu “ 0

are valid, taking Assumption 4.4 into account. Left-multiplying (C.13) by QJC yields

QJCARG

`

AJRe, t˘J

AJRze `QJCAVzjV “ 0 (C.15)

and subtraction from (C.13) leads to

ACC`

AJCe, t˘J

AJCze “ 0.

We obtain ze P im QC due to Lemma A.3. Left-multiplying (C.15) by zJe and taking(C.14) into account leads to ze P im QCRV. Hence we deduce that

ze P im QCRV

zjL “ 0

zjV P im QC´V

zjE “ ÁJEQCRVze

z"au“ 0

z"πu“ 0

zφu “ 0

hold true. We can choose a projector onto ker G1 py, tqJ and along im G1 py, tq by

WJ1 “

»

—

—

—

—

—

—

—

—

–

QCRV 0 0 0 0 0 00 0 0 0 0 0 00 0 QC´V 0 0 0 0

ÁJEQCRV 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

172


and

W1 “

»

—

—

—

—

—

—

—

—

–

QJCRV 0 0 ´QJ

CRVAE 0 0 00 0 0 0 0 0 00 0 QJ

C´V 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

respectively.


N1 py, tq “



`

AJCe, t˘´1

AVQC´VzjV ,


(

.

Proof . Let be z P ker G1 py, tq. This is valid if and only if

`

ACC`

AJCe, t˘

AJC ` ARG`

AJRe, t˘

AJRQC

˘

ze ` AVzjV “ 0 (C.16)


AJVQCze “ 0 (C.17)

zjE “ 0

ΛuAJEze “ z"πu

zφu “ 0

ΛuAJEQCze “ z"au

hold true, see (5.20), using Assumption 4.4, 3.31 and Property 3.32. Left-multiplying(C.16) by zJe QJ

C and taking (C.17) into account result in QCze P ker AJR due to Lemma A.3.We attain QCze P im QCRV using (C.17). With this we reduce (C.16) to

ACC`

AJCe, t˘

AJCze ` AVzjV “ 0 (C.18)


HC

`

AJCe, t˘

PCze ` AVzjV “ 0,

where HC

`

AJCe, t˘ “ ACC

`

AJCe, t˘


Left-multiplication of (C.18) by QJC leads to zjV P im QC´V. Hence we deduce that

QCze P im QCRV

zjV P im QC´V

173

zjE “ 0

zφu “ 0

PCze “ ´HC

`

AJCe, t˘´1

AVQC´VzjV


ΛuAJEze “ z"πu

ΛuAJEQCze “ z"au

are valid and in the end we achieve

N1 py, tq “



`

AJCe, t˘´1

AVQC´VzjV ,


(

.

174

Notation

AbbreviationODE ordinary differential equationDAE differential algebraic equationMNA modified nodal analysisIVP initial value problemBDF backward differentiation formulasKCL Kirchhoff’s current lawKVL Kirchhoff’s voltage lawME Maxwell’s equationsEM electromagneticMQS magnetoquasistaticPEC perfectly electric conductingFIT finite integration techniqueMGE Maxwell’s grid equationsI-cutset cutset of current sourcesLI-cutset cutset of inductors and current sourcesLEI-cutset cutset of inductors, EM devices and current sourcesV-loop loop of voltage sourcesCV-loop loop of capacitors and voltage sources

GeneralD there exists@ for allN naturalsZ integerR real numbersRn real n-dimensional vector spaceA P Rnˆm and A P Znˆm matrix with n rows and m columnsI identity matrixS set|S| number of elements in SdimS dimension of a vector space SSK orthogonal complement of S Ă Rn with respect to

the standard scalar product on Rn

I intervalD domain of definitionta1, . . . , anu set consisting of the elements a1, . . . , an

x PM x is an element of the set Mx RM x is not an element of the set MM Ă N M is contained in NM Ć N M is not contained in NMYN union of M and NMXN intersection of M and NM‘N direct sum of M and NMˆN product set of M and Nf : MÑ N function that maps from M into NB

Bxf partial derivative of f with respect to x

ddt

f total derivative of f with respect to t¨ function normCk pM,N q linear space of k-times pk ě 0q continuously differen-

tiable functions f : M Ñ N , M Ă Rm and N Ă Rn

openim A image of the matrix Aker A kernel of the matrix Arank A rank of the matrix Adet A determinant of the matrix AAJ transpose of the matrix AA´ pseudoinverse of the matrix AA` Moore-Penrose pseudoinverse of the matrix AA´1 inverse of the matrix AA´J transposed inverse of the matrix A~nK normal vector~nq tangential vector

Matrix Chain and SubspacesC1

d pI,Dq “ ty P C pI,Dq | d py p¨q , ¨q P C1 pI,RmquC1

D pI,Dq “ ty P C pI,Dq | D p¨q y p¨q P C1 pI,RmquD py, tq “ B

Byd py, tq

G0 py, tq “ A py, tqD py, tqB0 pz, y, tq “ B

ByrA py, tq z` b py, tqs

G1 pz, y, tq “ G0 py, tq ` B0 pz, y, tqQ0 py, tqN0 py, tq “ ker G0 py, tqS0 pz, y, tq “ tv P Rn|B0 pz, y, tq v P im G0 py, tquN1 pz, y, tq “ ker G1 pz, y, tqS1 pz, y, tq “ tv P Rn|B0 pz, y, tqP0 py, tq v P im G1 py, tquM0 ptq and H1 ptq obvious and hidden constraint setM1 ptq index-2 constraint set

ProjectorsQ0 py, tq projector onto ker G0 py, tq

P0 py, tq “ I´Q0 py, tqQ1 pz, y, tq projector onto ker G1 pz, y, tqP1 pz, y, tq “ I´Q1 pz, y, tqT pz, y, tq projector onto N0 py, tq X S0 pz, y, tqU pz, y, tq “ I´ T pz, y, tqW0 py, tq projector along im G0 py, tqW1 pz, y, tq projector along im G1 pz, y, tqR ptq projector onto im D py, tq and along ker A py, tq

Electric NetworkAR, AM, AL, AC, AE, AV, AI incidence matrix of elementse node potentialqM charges through the memristorsjM currents through the memristorsjL currents through the inductorsjV currents through the voltage sourcesjE currents through the electromagnetic devicesvs ptq given voltage sourcesis ptq given current sourcesqC pu, tq charges of capacitorsgR pu, tq currents of resistorsφL pj, tq fluxes of inductorsφM pq, tq fluxes of memristorsC pu, tq “ B

BuqC pu, tq

G pu, tq “ B

BugR pu, tq

L pj, tq “ B

BjφL pj, tq

M pq, tq “ B

BqφM pq, tq

Projectors for Electric NetworksQC projector onto ker AJCPC “ I´QC

QC´V projector onto ker QJCAV

PC´V “ I´QC´V

QVĆ projector onto ker AJVQC

QRMĆV projector onto ker“

AR AM

‰JQCQJ

C´V

QCRMV projector onto ker“

AC AV AR AM

‰J

PCRMV “ I´QCRMV

QRĆV projector onto ker AJRQCQJC´V

QCRV projector onto ker“

AC AV AR

‰J

PCRV “ I´QCRV

Electromagnetic Field~E electric field

~H magnetic field~D electric induction~B magnetic inductionρ distribution of charges~Jc conduction current density~Jd displacement current density~Jt total current densityε permittivityν reluctivityσ conductivityζ artificial material propertyξ artificial material propertyϕ scalar potential~A vector potential~Π auxiliary vector field

Discrete Electromagnetic Field"e, "eu (reduced) discrete electric field strength"

h,"

hu (reduced) discrete magnetic field strength""

d,""

du (reduced) discrete electric induction density""

b,""

bu (reduced) discrete magnetic induction densityq, qu (reduced) discrete distribution of charges density""

j c,""

j c,u (reduced) discrete conduction current density""

j t,""

j t,u (reduced) discrete total current densityMε, Mu

ε (reduced) discrete permittivity matrixMν , Mu

ν (reduced) discrete reluctivity matrixMσ, Mu

σ (reduced) discrete conductivity matrixMζ , Mu

ζ (reduced) discrete artificial material property matrixMξ, Mu

ξ (reduced) discrete artificial material property matrixφ, φu (reduced) discrete scalar potential"a, "au (reduced) discrete vector potential"π, "πu (reduced) discrete auxiliary vector

S, rS, Su, rSu (reduced) discrete divergence operators

G, rG, Gu, rGu (reduced) discrete gradient operators

C, rC, Cu, rCu (reduced) discrete curl operatorsΛu excitation matrix

Vector Analysis∇¨ divergence operator∇ gradient operator∇ˆ curl operator∆ Laplace operator∇2 vector Laplace operator

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189

Index

Aartificial material . . . . . . . . . . . . . . . . . . . . 56

discrete matrices . . . . . . . . . . . . . . . . .69reduced discrete matrices . . . . . . . . 80

auxiliaryvector . . . . . . . . . . . . . . . . . . . . . . . . . . . 83vector field . . . . . . . . . . . . . . . . . . . . . . 57

Bbasic elements . . . . . . . . . . . . . . . . . . . . . . . 93boundary excitation . . . . . . . . . . . . . . . . . 81Brauer’s model . . . . . . . . . . . . . . . . . . . . . . 53

Ccanonical solution set . . . . . . . . . . . . . . . . 14class of gauge conditions . . . . . . . . . . . . . 56

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 69reduced discrete . . . . . . . . . . . . . . . . . 80

conduction currentdiscrete . . . . . . . . . . . . . . . . . . . . . . . . . . 64

conduction current density. . . . . . . . . . .50conductive contact. . . . . . . . . . . . . . . . . . .59conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 52

matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 66consistent

initial value . . . . . . . . . . . . . . . . . . . . . .15initialization . . . . . . . . . . . . . . . . . . . . . 16

constitutive laws. . . . . . . . . . . . . . . . . . . . .52discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 66reduced discrete . . . . . . . . . . . . . . . . . 78

constraint sethidden constraint set . . . . . . . . . . . . 21index-2 constraint set . . . . . . . . . . . . 21obvious constraint set . . . . . . . . . . . .15

continuity equation . . . . . . . . . . . . . . . . . . 51discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 68

reduced discrete . . . . . . . . . . . . . . . . . 79cosimulation. . . . . . . . . . . . . . . . . . . . . . . .114Coulomb gauge . . . . . . . . . . . . . . . . . . 55, 57

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 70coupled

strongly coupled . . . . . . . . . . . . . . . . 114weakly coupled . . . . . . . . . . . . . . . . . 114

coupling condition . . . . . . . . . . . . . . . . . . 114cross talk. . . . . . . . . . .see proximity effectcurl operator

discrete . . . . . . . . . . . . . . . . . . . . . . 64, 67discrete reduced . . . . . . . . . . . . . . . . . 78

cutset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158I-cutset . . . . . . . . . . . . . . . . . . . . 100, 117LEI-cutset . . . . . . . . . . . . . . . . . . . . . . 117LI-cutset . . . . . . . . . . . . . . . . . . . . . . . 101

DDC solution . . . . . . . . . . . . . . . . . . . . . . . . 110deletion matrices . . . . . . . . . . . . . . . . . . . . 74differential-algebraic equation . . . . . . . . . 6

linear solution space . . . . . . . . . . . . . 14matrix chain . . . . . . . . . . . . . . . . . . . . . 16natural extension . . . . . . . . . . . . . . . . 15solution . . . . . . . . . . . . . . . . . . . . . . . . . 14

differentiation index . . . . . . . . . . . . . . . . . . 9displacement current density. . . . . . . . .51distribution of charges . . . . . . . . . . . . . . . 50

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 64divergence operator

discrete . . . . . . . . . . . . . . . . . . . . . . 64, 67discrete reduced . . . . . . . . . . . . . . . . . 79

drift-off-phenomenon . . . . . . . . . . . . . . . . 29

Eeddy currents. . . . . . . . . . . . . . . . . . . . . . .141

191

electric boundary condition . . . . . . . . . . 58electric field . . . . . . . . . . . . . . . . . . . . . . . . . 50

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 64electric induction . . . . . . . . . . . . . . . . . . . . 50

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 64electromagnetic device . . . . . . . . . . . . . . . 49electroquasistatic . . . . . . . . . . . . . . . . . . . . 53equivalent circuits . . . . . . . . . . . . . . . . . . . 93excitation matrix . . . . . . . . . . . . . . . . . . . . 81

pre- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Ffade-out projectors . . . . . . . . . . . . . . . . . . 73finite integration technique . . . . . . . . . . 60

Ggauge

condition . . . . . . . . . . . . . . . . . . . . . . . . 55freedom . . . . . . . . . . . . . . . . . . . . . . . . . 54function . . . . . . . . . . . . . . . . . . . . . . . . . 54invariant. . . . . . . . . . . . . . . . . . . . . . . . .55transformation. . . . . . . . . . . . . . . . . . .54

Gauss’ law . . . . . . . . . . . . . . . . . . . . . . . . . . 51discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Gauss’ law for magnetism . . . . . . . . . . . 51discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 63

grad-div regularization. . . . . . . . . . . . . . .69gradient operator

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 67graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

connected . . . . . . . . . . . . . . . . . . . . . . 158digraph. . . . . . . . . . . . . . . . . . . . . . . . .157directed . . . . . . . . . . . . . . . . see digraphedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157node . . . . . . . . . . . . . . . . . . . . . . . . . . . 157subgraph . . . . . . . . . . . . . . . . . . . . . . . 158undirected . . . . . . . . . . . . . . . . . . . . . . 157

Iincidence matrix. . . . . . . . . . . . . . . . . . . .158

reduced . . . . . . . . . . . . . . . . . . . . . . . . 159reference node . . . . . . . . . . . . . . . . . . 159

index . . . . . . . . . . . . . see tractability indexindex reduction . . . . . . . . . . . . . . . . . . . . . . 20index-2

index-2 components. . . . . . . . . . . . . .34index-2 constraint set see constraint

setindex-2 set . . . . see tractability index

initial value problems . . . . . . . . . . . . . . . . . 6internal boundary . . . . . . . . . . . . . . . . . . . 58

KKirchhoff’s

current law . . . . . . . . . . . . . . . . . . . . . . 97voltage law . . . . . . . . . . . . . . . . . . . . . . 97

Kronecker index . . . . . . . . . . . . . . . . . . . . . . 8

Lleading term. . . . . . . . . . . . . . . . . . . . . . . . . 13loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

CV-loop . . . . . . . . . . . . . . . . . . . . . . . . 101V-loop . . . . . . . . . . . . . . . . . . . . . . . . . 100

Lorenz gauge . . . . . . . . . . . . . . . . . . . . 55, 57discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Mmagnetic field . . . . . . . . . . . . . . . . . . . . . . . 50

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 64magnetic induction . . . . . . . . . . . . . . . . . . 50

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 64magnetoquasistatic . . . . . . . . . . . . . . . . . . 53magnetoquasistatic device . . . . . . . . . . . 57mass

contact . . . . . . . . . . . . . . . . . . . . . . . . . . 59node . . . . . . . . . . . . . see reference node

matrixpositive definite . . . . . . . . . . . . . . . . 149positive semidefinite . . . . . . . . . . . . 149

matrix pair . . . . . . . . . . . . . . . . . . . . . . . . . . . 8matrix pencil . . . . . . . . . . . . . . . . . . . . . . . . . 8

nonsingular . . . . . . . . . . . . . . . . . . . . . . . 8singular . . . . . . . . . . . . . . . . . . . . . . . . . . .8

Maxwell’s equations . . . . . . . . . . . . . . . . . 50Maxwell’s grid equations . . . . . . . . . 63, 83Maxwell’s house . . . . . . . . . . . . . . . . . . . . . 54Maxwell-Ampere’s law . . . . . . . . . . . . . . . 51

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 63discrete reduced . . . . . . . . . . . . . . . . . 78

Maxwell-Faraday’s law. . . . . . . . . . . . . . .51

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Index

discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 63memristance . . . . . . . . . . . . . . . . . . . . . . . . . 96memristor . . . . . . . . . . . . . . . . . . . . . . . . . . . 95modified nodal analysis . . . . . . . . . . . . . . 99monolithic. . . . . . . . . . . . . . . . . . . . . . . . . .114

Nnumerically qualified. . . . . . . . . . . . . . . . .17

OOhm’s law. . . . . . . . . . . . . . . . . . . . . . . . . . .52operating point . . . . . . . . . . . . . . . . . 15, 110

Ppath. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157

RV-path . . . . . . . . . . . . . . . . . . . . . . . . 110perfectly electric conducting see electric

boundary conditionpermeability . . . . . . . . . . . . . . . . . . . . . . . . . 52permittivity . . . . . . . . . . . . . . . . . . . . . . . . . 52

matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 66perturbation index . . . . . . . . . . . . . . . . . . 10phantom objects . . . . . . . . . . . . . . . . . . . . .71pill-box method . . . . . . . . . . . . . . . . . . . . . 58point of equilibrium . . . . . . . . . . . . . . . . . 43projector . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

along . . . . . . . . . . . . . . . . . . . . . . . . . . . 150complementary . . . . . . . . . . . . . . . . . 151onto . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150orthogonal. . . . . . . . . . . . . . . . . . . . . .150

properly stated leading term . . . . . . . . . 13proximity effect . . . . . . . . . . . . . . . . . . . . 140pseudoinverse . . . . . . . . . . . . . . . . . . . . . . 153

Moore-Penrose . . . . . . . . . . . . . . . . . 155

Rreduced curl-curl equation . . . . . . . . . . . 82reduced incidence matrix. .see incidence

matrixreference node . . . . . see incidence matrixreluctivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Sscalar potential . . . . . . . . . . . . . . . . . . . . . . 54


selection matricesboundary . . . . . . . . . . . . . . . . . . . . . . . . 77unknown . . . . . . . . . . . . . . . . . . . . . . . . 77

shrinking matrices see deletion matricesskin effect . . . . . . . . . . . . . . . . . . . . . . . . . . 141strangeness index . . . . . . . . . . . . . . . . . . . . 12

Ttotal current density . . . . . . . . . . . . . . . . . 51

reduced discrete . . . . . . . . . . . . . . . . . 81tractability index . . . . . . . . . . . . . . . . . . . . 17

index-0 set. . . . . . . . . . . . . . . . . . . . . . .17index-1 set. . . . . . . . . . . . . . . . . . . . . . .17index-2 set. . . . . . . . . . . . . . . . . . . . . . .17

tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Vvector potential. . . . . . . . . . . . . . . . . . . . . .54


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Coupled Electromagnetic Field/Circuit Simulation: Modeling ... · g angige Methode in der Praxis...

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Transcript of Coupled Electromagnetic Field/Circuit Simulation: Modeling ... · g angige Methode in der Praxis...